Painting - bathroom walls

I have a problem with the paint peeling on the walls and ceiling within the shower area of my bathroom. Is there a paint that will resist the heat and moisture best?

Eggshell finish paint is a good choice.

I would address the peeling problem first. What exactly is the paint peeling down TO? Old paint? Plaster? Drywall? Many times, paints fails from a previous coat because a latex was applied to an exisiting oil without priming or proper preparation. If it is oil, you'll prime first or repaint with oil after a good priming. Check for oil-based paints by putting a liberal amount of acetone-based nail polish remover on a piece of cloth and rubbing small circles over the old paint for about 20 seconds....either the color is transferred to the cloth, or it is not. If so, it was latex. If not, 'twas oil. If it is plaster, scrape and sand, clean, then prime with a good oil primer and paint with latex. There are mildew-resistant latex paints such as 'Permawhite' by Zinsser that will prohibit mildew growth in humid areas if required. If drywall, prime with latex, paint of your choice on top. At any rate, the area must be scraped and feather sanded, wiped with a damp rag to remove the dust, then properly primed and painted. Oil-based paints inherently resist moisture better than acrylics/latex paints as they are less permeable to water vapour than are latex products. In your case, I suspect that whatever your choice for topcoat is, that the failing substrate, once addressed properly, is the focus of your troubles and once eliminated, you'll be pleased. Excuse my long windedness, but I get rather particular when it comes to coatings failure. :D

This is a condo, ceiling and walls are drywall. Latex paint was used, but no primer. Peeling paintis right down to the drywall.
Looks like builder took a cheap way out? Help!

Right then...as far as 'resisting' moisture goes, oil is the superior film. If you're talking mildew resistance, a latex is best, such as Permawhite. Sand and wash the area, allow to dry well. Prime with a good acrylic, then paint with semigloss oil paint.