Check to make sure the bow fits the hand; the thumb should at least be close to touching the index finger when the hand is closed around the grip, but thumb and index finger should not overlap more than an inch or two (if the archer's fingers are actually touching their palm, the bow's grip is likely too small). Also make sure the bow is not too heavy, physically, for the archer it is intended for. If the bow does not fit the hand of the person using it, or it feels "heavy" while being used -- if the archer is not comfortable while using the bow, then it (the bow) will never be the "right" bow for that person.
Make sure the bow is the right length. The bottom limb tip should not touch the floor (or ground) when the bow is being used, nor should the bow be so short that the bowstring is at such an acute angle (in relation to the arrow) at full-draw that the archer's fingers are severely pinched.
If it's a used bow, look down the length of the bow and see if the limbs are straight and in-line with each other; if the limbs are twisted too much, they can not be straightened, and the bow should not be used. The limbs should be checked for twisting both with and without the string installed, because limbs which appear straight without the string installed can sometimes become twisted with the string installed (and the reverse is also true). The important thing is that the limbs are not twisted when the string is installed and the bow is drawn back; with twisted limbs, there is a real danger of the string coming out of the string nocks on the bow.
Check the string for signs of wear. The loops should be "clean" (without fuzz), and the serving (the wrapping of thread around the loops) must be intact; a slight amount of fuzz is permisible, but should be watched to make sure it doesn't become broken strands. The nocking area (where the arrow nock sits on the string) should also be "clean", as well as relatively firm and solid-feeling (compared to the un-served portion of the string), and the serving should be intact. A slight amount of fuzz on the string can be taken care of with wax, but watched closely so it doesn't become broken strands. If there are any broken strands anywhere on the string, the string needs to be replaced; the most likely places the string will break are in the loops, under the nocking-point serving and at the ends of the nocking-point serving. If the string has any knots tied into it, it should be replaced; tying knots into the string only weaken the string where the knot is tied.
Also look for cracks of any kind going across the limbs (transverse cracks); the best way to do this is to bend each limb by hand, while at the same time watching closely for any "invisible" cracks to show themselves. The limbs should be crack-free, although longitudinal cracks (those going along the length of the limb) are sometimes not a danger, so long as they do not extend into the limb tip, where the string nock is; if there are any longitudinal cracks in the bow's limbs, take the bow to a professional to have it looked at. A bow with transverse cracks which go deeper than the paint or varnish on the bow is already broken, and should not be used. If there is any doubt as to how deep a crack is, or if it might be a danger to the bow, always take the bow to a professional to have it looked at.
It also depends on what the bow will be used for. If you want something to take down an elk, make sure it has a sufficiently high draw-weight (at least 50lbs for elk and larger bears; at least 40lbs for white-tail deer and other animals), while for target work a lower-poundage bow is normally sufficient. A heavily recurved bow shoots faster, but also tends to be less "forgiving" of mistakes; faster bows are often harder to control because they tend to have more of what is known as "hand shock", where "excess" power is transmitted into the hand.
Also to consider are such things as: does the archer want a take-down model for convenience , or a one-piece model for beauty and traditionalism? Does the archer want a modern recurve such as a Bear Kodiak, or an older design such as a Magyar Horse Bow? Does the archer want a wood, laminate, or metal riser (the part of the bow which comprises the handle, between the limbs)? Does the archer want to use accessories such as a bolt-on arrow rest, or perhaps sights? Some of these considerations are purely cosmetic -- i.e., up to the aesthetic preferences of the archer, while others are more practical in nature.
There may be other things, not included here, which an archer would want to look for in a recurve bow. In such cases it is probably best to find a reputable Archery store which deals in Traditional Archery, and ask the professional(s) there for help.