Table-driven lexers in Engineering a Compiler 2nd Edition
Let me answer your questions very quickly:


NextChar(char) reads a character and assigns it to the variable char.
The new character is appended to the current lexeme being built via the command lexeme = lexeme + char.
The table Delta implements the transition function $\delta$ of a DFA (sometimes called an FSM). There is a lot of information on DFAs on the web and even on Computer Science.


I was able to answer these questions via a combination of common sense and a very rudimentary knowledge of how lexers work. More generally, when faced with code or pseudocode which you don't quite understand, you should try to figure out what the code is trying to accomplish, and how. For example, NextChar surely reads a character. You might expect it to return it, but from the context it is clear that it is actually stored in its argument. The fact that it is adjoined to lexeme significantly strengthens this guess.