Before we start...
Never apply final varnish to an oil painting unless it has thoroughly dried. Oil paint requires anywhere from 6 to 12 months (or longer) to dry depending on how thick the paint was applied to the canvas. If you want to add a temporary finish to paintings that have not dried thoroughly you should consider using a retouch varnish.
If you have large sunken, i.e. dull and flat, areas on your original unvarnished oil painting you might want to consider oiling out the painting before applying a final varnish. Varnishing will even out the finish on paintings that are fairly even already, but if you have areas that are high gloss along with areas that are clearly sunken, "due to variations in the types of pigments and/or mediums used" , then applying a final varnish may not even out the finish as you might be expecting. Oiling out is suggested in such a case. If you decide to oil out your painting you will need to wait an additional 6 months before applying your final varnish and it's best to perform the oiling out process shortly after your painting has been completed, within the first few weeks.
In this tutorial we will accomplish the following: 1) learn a method for applying multiple thin coats of varnish, 2) learn a brushing technique that will minimize uneven brush strokes that could be visible in the final product, and 3) achieve a high gloss finish similar to many paintings that are displayed in museums, sometimes referred to as a 'museum finish'.
Additional points to remember when varnishing:
Use Good Ventilation: You will notice in the images used throughout this tutorial that I am working next to a window and I am using a fan to keep the air moving. Varnishing per this tutorial can take from one to two days so be sure that your work area is properly ventilated to prevent fumes from building up.
Brush Selection: The largest painting that is varnished in this tutorial is 12"x16" and the smallest is 4"x6". I chose a fine bristled brush that is 1" wide. For paintings larger than 12"x16" you should use a brush with a width in the 1-1/2" to 3" range. Don't be cheap when purchasing a varnishing brush unless you want to be picking bristles out of your varnish. Also, once you use a brush for varnishing DO NOT use it for painting, and if you have a brush that you previously used for painting DO NOT use it for varnishing. Purchase a varnishing brush that will only be used for this purpose.
Winsor & Newton Dammar Varnish: "You will need some Dammar varnish thinned 50% with turpentine. Turpenoid or mineral spirits will not dissolve the Dammar; it must be real turpentine."  For this tutorial I used Winsor & Newton Dammar Varnish that comes pre-thinned (50% Dammar / 50% real turpentine). It is ready to be applied from the jar without further thinning. DO NOT shake your jar of Dammar because this could cause air bubbles to form in the varnish which is something you want to avoid. We are now ready to start...
STEP BY STEP DAMMAR VARNISH
Assemble all materials and tools before starting.
Heat some water until almost boiling, pour some of the hot water into the first ceramic bowl (item 7a).
Fill the ceramic bowl with enough water so that the jar of Dammar Varnish (item 6) sits mostly submerged in the water. Let the Dammar warm for about 5-10 minutes. Warming the Dammar will lower its viscosity without the need to add more turpentine, making it easier to apply. I recommend using a ceramic bowl because of the heat retaining properties of ceramic - you want to keep the Dammar warm until you have finished applying a thin coat of varnish to your painting(s).
I have seen another tutorial using Winsor & Newton Dammar Varnish where the warming process I describe in this tutorial was not used. In that tutorial the Dammar was poured directly into a pot (without warming) and then applied directly onto the canvas with a brush. The warming procedure that I describe in this tutorial seems fairly common in other varnishing tutorials. With that said... there are some words of caution: "Varnish is highly flammable, so electric heating / warming devices are not recommended while using varnish!"  The hot water method seems to be a widely accepted and safe alternative.
Before removing the jar of Dammar that has been warming in the first ceramic bowl, pour fresh hot water (not boiling) into the second ceramic bowl. (item 7b)
Insert the glass pot (item 8) into the ceramic bowl with the hot water, ensuring that the glass pot is sufficiently submerged under the water.
You can use a different style container then the ones I am using as long as the pot fits into the warming bowl without displacing to much hot water, and it has a mouth large enough to accept your varnishing brush. (See reference photo below that shows another artists setup)
Pour some of the warmed Dammar Varnish from the jar (item 6) into the glass pot (item 8) that is sitting in the warming bowl (item 7b):
In this tutorial I used a 75ml jar of Winsor & Newton Dammar Varnish and I poured approximately 37ml into the glass pot. As a general rule have at least 1/2" of varnish in your pot. "Every time I add more varnish to the pot, I also change the hot water that the jar warms in (step 2), and the hot water that the pot sits in (step 3), to maintain the varnish temperature." 
Dip your brush (item 4) into the heated Dammar and lightly wipe each side of the loaded brush against the side of the glass pot (item 8) to remove excess Dammar:
If the brush is overloaded with Dammar and you begin brushing it onto the canvas then you might develop a puddle of varnish on your canvas and your coating will likely be to thick. On the other hand, if you remove to much varnish from the brush you will not get enough coverage, your brush will run dry before you can make one pass across the canvas.
Using the paper towel (item 2), touch the tips of the bristles to remove excess Dammar from the tip of the brush (item 4):
You might think that removing all this excess Dammar from the brush will leave us with to little varnish to cover the canvas. Our goal in steps 5 and 6 is to remove excess varnish and prevent puddling of varnish on the canvas and prevent applying to thick of a coating of varnish at one time. At the same time we want to have enough varnish in the brush so that we do not run dry half way across the canvas as we are moving the brush from one side of the painting to the other. The technique we are trying to develop in this tutorial is the application of multiple thin coats of varnish.
With a little practice you will learn how much excess Dammar to remove from your brush to prevent either of these situations.
Before we start applying varnish let me explain the process briefly. We will be applying varnish first in columns (along the vertical axis of the painting) and than rows (along the horizontal axis of the painting). We will brush on the varnish one column at a time, first in one direction then back over the same brush stroke in the opposite direction. Once we cover the entire painting this way, column by column, reloading the brush with each back and forth pass, we will rotate the painting 90 degrees and repeat the brush strokes in the same back and forth pattern along the horizontal axis, row by row, but this time without reloading the brush with fresh varnish after each pass.
I have listed Winsor & Newton's recommended application method for their Dammar Varnish: 
- Apply the varnish in 1-3 thin coats, rather than 1 thick coat. A thick coat will take longer to dry, may dry cloudy, drip or sag during application and has a greater chance of showing brush strokes when dry.
- Thinned varnish is more susceptible to producing bubbles. Do not be vigorous in your application.
- Apply in long even strokes to cover the surface top to bottom while moving from one side to the other. While working, inspect the varnish layer at all angles for bubbles. Even them out immediately.
- Once you leave an area, do not go back over areas that you have done. If you do, you risk dragging partially dry resin into wet, which will dry cloudy over dark colors. If any areas were missed, allow to dry completely and re-varnish.
Starting at one of the corners of your canvas, place your brush into position and in one long brush stroke across the painting. Make one long even brush stroke from one side to the other side and then leave it alone. Do not stop part way through. Do not go back to touch up a spot you feel could be better unless you see a drastic problem like large areas of no varnish or bubbles.
Reminder: Even though we have warmed the Dammar, we need to work quickly because once applied to the canvas the Dammar will begin setting immediately. After about 10 minutes the varnish will be tack dry.
After reloading the brush make another long stroke in the opposite direction, directly over your first brush stroke. As we did with the first brush stroke, move the brush in one long even stroke from one side to the other side. (Note: When varnishing landscape I always apply my first strokes along the vertical plain. When I rotate the canvas to make a pass over the first coating of varnish I like to be moving the brush along the horizontal plain.)
It is a good idea to make a brief scan of your progress after each back and forth pass of the brush to ensure your getting even coverage. If you are seeing large uncovered areas as you move across the painting than you are probably removing to much varnish from the brush during steps 5 and 6.
However, if everything is going well, you should start seeing the difference that the varnish is having on the oil paintings appearance. Notice how the colors are popping, as if they were just freshly painted. (see step 9 image)
Repeat steps 5 through 8 until you have completely coated the painting with varnish. When starting a new row you should slight overlap your new strokes with the previous strokes. The next step (10) is very important for evening out the coating of varnish that you just applied.
Rotate your painting so that you will be working in the horizontal plain. Without reloading your brush start at one corner and brush straight across the painting without stopping and then brush back over your stroke in the opposite direction. Repeat this until you have gone over the entire painting. This step will help to distribute the Dammar more evenly across your painting.
At any point during the application of the Dammar Varnish you might find a piece of debris has fallen onto the canvas. Using the tweezers (item 10), gently remove the contaminant while trying not to disturb the varnish.
Once you have completed coating your painting with varnish place your brush into a container (item 9)with turpentine. This will keep the varnish from hardening onto your brush while you wait for the varnished paintings to dry enough for you to continue with the second, and third coatings of varnish.
Varnish is reusable so you should pour, using a funnel (item 12), whatever you have not used back into its original jar (item 6). I recommend doing this between varnish coats. It is recommended to let the applied varnish dry for a minimum of two hours between coats, although the time might vary depending on the temperature and humidity in your area. If the varnish is tacky to the touch it's not ready for a second or third coating. I test my varnish by pressing a finger along the edge of the canvas, if I see a finger print its not ready. One source I spoke with said you could wait a full day between varnish coatings.
If you are going to apply two or more coats of varnish please start this tutorial from the beginning, working each new coat of varnish as if it was the first. This would mean starting again at step 2.
After about 10 minutes the varnish should be tacky enough to not run or drip. Kate is here holding up a painting after she made an examination of the varnish. You can see problem spots by holding the painting up to a light source and looking across the surface at almost eye level with surface of the canvas. Missed spots or uneven brush strokes will be very visible. Its during this examination that you will decide if another coat of varnish is needed. "If needed, apply a second [or third coat] to fix rough areas after they dry. Do not try to fix areas by themselves, but rather apply a complete second [or third coat] of the varnish."  I have found that three coats of varnish gives me the results I am aiming for.
Here I am varnishing another painting. I like to have several paintings ready to go before I start a varnishing session. Varnishing a painting and applying two to three coats of varnish can take one to two days, and so I like to have several paintings being varnished at the same time. As I finish varnishing one painting I move it aside and start my second, and third, and so on. I than let them all dry together.
- Jamie Williams Grossman, www.wetcanvas.com, How to Varnish an Oil Painting.
- David Pyle, www.winsornewton.com, The Last Layer: The Why, What, When, and How of Varnishing.
- Jeff Beer, How to Simply Varnish an Oil Painting.