How can I give players custom names?

This depends on whether you want to change the name entirely or not.
If so, then no you cannot. However if you only want to add a prefix or suffix, then yes this is possible.
In order to do so, you can create a team using /team create <name> and then modify the team using /team modify <name> <prefix/suffix> <text>. Then add the person to that team using /team join <name> <player>. Doing this you can also make the name coloured.