As has been mentioned many times, there are four kinds of memory--tactile, aural, visual, and analytical. The tactile or touch memory is purely mechanical, that is, the sensation of the fingers playing through the piece. Aural memory is the music being retained for recall in the "mind's ear". Visual is literally "photographic memory". There are people who have that uncanny ability, and they can actually view of the sheet music in their mind as if it were on the music desk before them. Analytical memory arises from studying the piece away from the piano, noting its form and landmarks such as parts, key modulations, figuration patterns, cadences, transitions, climax, unusual harmonies, etc. Some people are more adept at music theory than others and can use this approach expertly. Geiseking was a genius at this.
Generally, the trick seems to be to use any TWO of these methods with which you are most comfortable. NEVER depend on just one, particularly tactile memory. If the fingers suddenly lose their place, the piece will come to a abrupt halt! If, however, you are combining aural memory with tactile, for example, then the mind's ear can take over momentarily to help. For example, it will assist you in imrprovising a bit through a difficult measure or two to preserve the overall effect. Or it can help you to advance to a nearby landmark where you can regain your bearings, while maintaining continuity. The rule there is never stop--keep on going, even if you must momentarily become a co-composer with Chopin or whomever. (Don't worry--if Chopin is looking down on you, like any other composer, he'll be totally intrigued by your improvisation during that emergency.)
I always found that if I truly loved a piece, I could memorize it. Incidentally, as you get older, memorizing becomes more and more difficult. (The brain's RAM becomes too full basically.) And I fall into that category! Yet, recently I was playing the accompaniment to "Believe it not!", a powerful, tempestuous, and sensual song by Rachmaninoff. After practicing it for awhile, I closed the music and astonished myself by playing it through in its entirety--I had memorized it because I really loved the piece, and had the tactile and aural sensations clearly ingrained into my brain.
I think there's a larger message there that transcends mere memorization. Once you become a "well-rounded pianist", do not spend time on repertoire you dislike. The repertoire is too vast and one's lifetime is too short. So select music you truly love. Think of a piece as an important intellectual and emotional investment.
Be sure to play the piece(s) for relatives and friends before you get before an audition, jury, or audience. Be secure in your playing before taking the plunge.