The Lost City is essentially a point-and-touch adventure game. In it you are returning to an island in search of the lost city in order to return an artifact that your grandmother passed on to you. Each location is essentially a static picture with minimal graphical effects such as perhaps a little running water or rain or a glowing orb. You navigate by tapping on the edges of each frame (i.e. tap the bottom to go backwards) or on things within the frame (i.e. a path leading off into the distance). As you do so a map is weaved together that you have access to via your backpack. Your backpack also contains a journal so as you discover things of interest (and the game auto adds such things to your backpack so you'll KNOW it's of interest) they are scribed for later viewing.
As for the puzzles, many of them rely on recognizing that this shape I found over here fits a puzzle over there. Or this object makes sense if it's use on that. The game requires good observational skills so that you don't miss anything in a location and reasonable skills at putting things together. On a personal note, I found the game to be pretty much perfectly attuned to my skill level whereas I recall Myst feeling a little too difficult for my lack of patience. Generally speaking, if I'm stuck on something for 5 minutes in a game like this then I move on to something else, however, The Lost City actually includes a built in hint system to prevent that from happening.