A parser is generally generated from the grammar. See Language - Compiler compilers or (lexer|parser) generators
Parsing is done generally at the token level but can be done at the character level when the lexer and parser are done in one step: See wiki/Scannerless parsing
Syntax analysis is also known as Sentence recognition
Additional step can be added to the parse phase in order to construct an Abstract Syntax Tree (AST) from the parse tree.
The term parsing comes from Latin pars (orationis), meaning part (of speech)
A syntax analyzer would check: