What are terminal and nonterminal symbols?

I am reading Rebol Wikipedia page.

"Parse expressions are written in the parse dialect, which, like the do dialect, is an expression-oriented sublanguage of the data exchange dialect. Unlike the do dialect, the parse dialect uses keywords representing operators and the most important nonterminals"

Can you explain what are terminals and nonterminals? I have read a lot about grammars, but did not understand what they mean. Here is another link where this words are used very often.

Definitions of terminal and non-terminal symbols are not Parse-specific, but are concerned with grammars in general. Things like this wiki page or intro in Grune's book explain them quite well. OTOH, if you're interested in how Red Parse works and yearn for simple examples and guidance, I suggest to drop by our dedicated chat room.

"parsing" has slightly different meanings, but the one I prefer is conversion of linear structure (string of symbols, in a broad sense) to a hierarchical structure (derivation tree) via a formal recipe (grammar), or checking if a given string has a tree-like structure specified by a grammar (i.e. if "string" belongs to a "language").

All symbols in a string are terminals, in a sense that tree derivation "terminates" on them (i.e. they are leaves in a tree). Non-terminals, in turn, are a form of abstraction that is used in grammar rules - they group terminals and non-terminals together (i.e. they are nodes in a tree).

For example, in the following Parse grammar:

greeting: ['hi | 'hello | 'howdy]
person:   [name surname]
name:     ['john | 'jane]
surname:  ['doe | 'smith]
sentence: [greeting person] 

• greeting, person, name, surname and sentence are non-terminals (because they never actually appear in the linear input sequence, only in grammar rules);
• hi, hello, howdy with john, jane, doe and smith are terminals (because parser cannot "expand" them into a set of terminals and non-terminals as it does with non-terminals, hence it "terminates" by reaching the bottom).

>> parse [hi jane doe] sentence
== true
>> parse [howdy john smith] sentence
== true
>> parse [wazzup bubba ?] sentence
== false

As you can see, terminal and non-terminal are disjoint sets, i.e. a symbol can be either in one of them, but not in both; moreso, inside grammar rules, only non-terminals can be written on the left side.

One grammar can match different strings, and one string can be matched by different grammars (in the example above, it could be [greeting name surname], or [exclamation 2 noun], or even [some noun], provided that exclamation and noun non-terminals are defined).

And, as usual, one picture is worth a thousand words:

Hope that helps.