Such grammars are termed context free grammars. A language is context free if and only if it can be characterised by a context free non-regular grammar. Context free grammars are equivalent to pushdown automata. Such grammars are termed regular or linear grammars. A language is regular if and only if it can be characterised by a regular grammar.
Regular grammars are equivalent to finite state automata.
A finite state automaton accepts all regular languages. A distinction is made between deterministic finite state automata and nondeterministic finite state automata. Determinism refers to the choices which an automaton has in order to make the next move. A finite state automaton is said to be deterministic if, during recognition, its behaviour is completely determined by the state it is in and the symbol it is looking at. The nondeterministic finite automaton, on the other hand, may have several possible next moves.