Mouthfeel on a BJCP Scoresheet is only about 10% of the score you assign to a beer (5 points out of 50), but there are things you can sometimes feel with your mouth that might not be as obvious to your other senses. There are several things that a judge should look for. The scoresheet lists the basics of body, carbonation, warmth, creaminess, and astringency, and then the catch all "other palate sensations". Let's deal with the basics first.
Body and carbonation are well-defined in each style. Ranges of body can go from very light to very heavy as descriptors. Think of the difference between Bud Lite and a viscous Russian Imperial Stout. It's the difference between water-like and downright chewy. To some degree, carbonation will play into this in that a light bodied beer with effervescent carbonation will be pleasantly refreshing on a summer day compared to a winter warmer where a thick mouthfeel and moderate low carbonation can be more desirable.
Warmth is then an important factor in mouthfeel as well. It tends to come from alcohol, so in most cases it should either be there or not. Lower alcohol beers should have no perceptible warmth from alcohol since there simply isn't that much present. Most people will be able to perceive some degree of alcohol after about 7-8%. Once you start to hit 10%, it will become more obvious. In nearly all cases, though, the alcohol should come across as a pleasant warmth, and not a hot sensation. Typically a hot, burning alcohol sensation is from the higher alcohols, also called fusel alcohols. These will create an almost solventy sensation. If you've ever tasted pure grain alcohol or even some cheap vodkas and whiskeys (where the alcohol has not had time to age and smooth out), you'll understand why this isn't a pleasant or desirable thing.
Creaminess, on the other hand, is a very desirable thing for some beer styles. It can be described as a soft, coating sensation most often associated with oatmeal stouts. There are several ways to achieve it, but generally most of them involve increasing the non-fermentable sugars in the beer. A higher temperature in the mash will generally convert more starches to non-fermentable sugars, so mashing at the higher end of the scale (say 155-158F) will get you a higher mouthfeel. Another common technique is an unmalted grain such as oatmeal or flaked barley which will give your beer a silky mouthfeel when added to a mash. You want to keep it under 5 or 10% though since it will affect the consistency of your mash and potentially make your sparge (where the wort is drained from the grains) tougher. Other techniques include carbonating with a Nitrogen gas mix or adding Lactose, which is another non-fermentable sugar derived from milk (hence the name of Milk Stouts).
Astringency is the opposite of creaminess. It's a harsh drying sensation similar to sucking on a teabag. There are a few common causes. Oversparging is trying to extract every last bit of sugar from the grain by continuing to rinse and drain the grain after most/all of the sugar has already been extracted. This could lead to tannins being extracted from the grain which will cause an astringent bitterness in the beer. By that same token, water with a relatively high pH (generally above 6.0) could also do the same. A general rule of thumb is to keep your mash at a pH of 5.2-5.6 and keep your sparge water below 6.0. It's more an issue with darker and roasted grains since they have more tannins (like tea and coffee) available to be extracted. Other possible causes are sparging with water that is too hot (temperature recommendations vary, but higher than the low 170s should be avoided) or bacterial infections (namely the vinegar tone producing aceto bacteria).
Finally, other palate sensations include anything else. One of the most common flaws in beer is diacetyl. Some describe its aroma as butter or butterscotch. For me, it comes off like movie popcorn butter. However, butterscotch can also be a description applied to kettle caramelization (when sugars in wort are boiled for a long time, they can caramelize). The tell tale difference is mouthfeel. Diacetyl has a slick, oily, and sometimes soapy feel that many will detect even if they don't get the aroma. Also, since soaps generally have a fatty lipid component to them, beers which wind up having a high lipid content will feel soapy. The cause is generally similar to autolysis where the beer is left on the yeast too long, and the yeast starts to eat fatty components in the trub.
In summary, although it's a relatively small component on a score sheet, mouthfeel can be very important in informing your other senses (similar to appearance) about what may be wrong or right with a beer. Cheers!