As a demonstration, take your right arm and bend it at a slight angle from being straight, then push down on it with your other arm and resist this motion. You will see that the primary muscle engaged are your forearm nearest your bicep (often called the 'blade' of your forearm), and your bicep muscles. So, probably the most effective exercises you can do for that resistance are hammer curls and close-grip palms-out pullups. Both will work your upper forearm and your lower bicep muscle.
Dumbell Hammer curls - Do one arm at a time. While standing, take a dumbell in your hand, let your arm rest at your side, with your palm facing directly into your body. Then bend your arm straight up toward the sky, without moving your elbow. Pause briefly at the top and squeeze the muscles, then slowly go back down. If you look in the mirror, it kind of looks like you are using a hammer (although usually you are pushing a hammer down primarly instead of pulling it up). Make sure to maintain good form by keeping your upper arm pointing straight toward the ground and not letting your elbow move around. You may have to use very little weight to accomplish this at first, but form is everything. Focus so that all of the power you are using to move the dumbell is in your upper forearm and biceps.
EZ-bar or Barbell Reverse Bicep Curls - This is performed much like the hammer curls, but using both hands at once. Depending on how wide or tight your grip is, you will focus less or more on your forearm muscles, respectively. It is usually impossible to do this motion without moving your elbows because your arms will probably be at angles toward your chest, so focus on MINIMIZING elbow movement. Again, use as low of weights as necessary to achieve proper form.
Close-grip palms-out pullups - On a pullup bar, assume a close grip, hands about 1-1.5 fists' width apart (imagine a third fist in between your actual fists for spacing). Perform pullups starting from a low-hanging position. The trick to this is to focus and keep your lifting power in your forearms, rather than in your shoulders or back. You will inevitably use some power from your back, but you should find it much less strenuous to your back than doing a wide-grip pullup.
In any given workout it does not make sense to do more than 1 or 2 of these exercises along with your other bicep exercises, because they are fairly redundant considering what a small percentage of muscle you are targetting.
In addition you may find it helpful to strengthen your lower forearm muscles with forearm exercises such as palms-up dumbell wrist curls, palms-down dumbell wrist curls, and so forth.
IMPORTANT: Also note that in an actual situation in which you are being arm barred, you will not ONLY use your forearm blade and biceps; you will also be using your shoulder (probably mostly your front deltoids) to try to torque your arm toward your body, your outer chest to do the same (you can hit that part of your chest with dumbell/machine/cable flyes), and your entire core (hips, lower back, abs) to torque all of your weight opposing the arm bar. Quickly you can begin to see that it is a little silly to only lift in order to counter one particular move, and that your best bet is to use a wide variety of exercises to increase your strength all over.
