This video explains the different types of cue tips available. They range from very soft to very hard with all the variations in between.
Basically, a soft tip will contact the cue ball a little softer and give you more feel on the shot. A hard tip will last longer before mushrooming and will provide more power transfer from the cue stick to the cue ball, which is why it is a good choice for a break stick.
The cue tip should be shaped to a crown shape as shown and roughened up so it holds the chalk better. Chalk should be used before every shot for the best cue tip performance.