Regular vs Context Free Grammars

Regular grammar is either right or left linear, whereas context free grammar is basically any combination of terminals and non-terminals. Hence you can see that regular grammar is a subset of context-free grammar.

So for a palindrome for instance, is of the form,

S->ABA
A->something
B->something

You can clearly see that palindromes cannot be expressed in regular grammar since it needs to be either right or left linear and as such cannot have a non-terminal on both side.

Since regular grammars are non-ambiguous, there is only one production rule for a given non-terminal, whereas there can be more than one in the case of a context-free grammar.

I think what you want to think about are the various pumping lemmata. A regular language can be recognized by a finite automaton. A context-free language requires a stack, and a context sensitive language requires two stacks (which is equivalent to saying it requires a full Turing machine.)

So, if we think about the pumping lemma for regular languages, what it says, essentially, is that any regular language can be broken down into three pieces, x, y, and z, where all instances of the language are in xy*z (where * is Kleene repetition, ie, 0 or more copies of y.) You basically have one "nonterminal" that can be expanded.

Now, what about context-free languages? There's an analogous pumping lemma for context-free languages that breaks the strings in the language into five parts, uvxyz, and where all instances of the language are in uvⁱxyⁱz, for i ≥ 0. Now, you have two "nonterminals" that can be replicated, or pumped, as long as you have the same number.