Play for yourself if they don't listen
Print a list of tunes you know and like to play. If you find yourself in a rut playing the same twenty tunes every night, refer to the list.
What fits depends on the venue. There's a difference between playing in a fine dining establishment and playing in a biker bar. Assuming a fine dining environment, what fits is anything that doesn't blow them out of their chairs, and not too many up tempo tunes. I do mostly ballads and medium tempo tunes. I toss in an occasional stride piece just to break the monotony.
Make eye contact, welcome them as they come in. If they are seated withing speaking distance or if you have a vocal microphone introduce yourself and solicit requests. If you don't get their attention, don't worry about it. You are organic Muzak.
Try to have the piano positioned where they have to walk past it on the way out. Put the tip jar in a prominent location. Put a couple bucks in it before you start. Bait.
Make sure to acknowledge special events such as birthdays and anniversaries. The waiters and waitresses usually know about such things. Ask them to tell you.
When a bartender or waiter/waitress helps you out, it's customary to give them a couple bucks from your tip jar at the end of the evening. If it's a steady gig, you'll be paid back a hundred fold in service and favors.
I keep a list of tunes sorted by the year they were on the pop charts. The list goes back to the mid-fifties and it sits unobtrusively on the piano. It's not all the tunes that were on the charts, only the romantic ones I know. When a couple is seated nearby and seem to be paying some attention, I ask them what year they first got together. Then I play (and sing, if it's a singing venue) a couple of tunes from that year mentioning the original artist so they are sure to recognize it. It's an old device that I first saw used by John Davidson, and it really works. Other people mention their years and it gets some audience participation going. This works well at a piano bar, too. It's very effective with senior citizens.
You can build your own list from the tunes listed at
http://www.webfitz.com/lyrics/lyricsbb.html.
Al Stevens