There are a number of words and phrases that have similar meanings to fear. They aren't exact synonyms though; there is always a small difference in effect or connotation.

Terror is extreme fear, as well as the use of fear to control or influence people.

Horror is related to fear but also suggests shock and sometimes even disgust.

Anxiety is a more general low-level fear, related to unease rather than terror.

Nervousness is usually a feeling of unease, often about a particular situation - it's not quite as strong as fear.

Fright is quite close to fear in its intensity, but it's more specific - a fright is basiclally a brief, scary shock (ie 'to give someone a fright').

Panic is overwhelming fear, stronger even than terror - it comes from the ancient wood-god Pan, and originally meant the kind of fear you might feel in the presence of an unknown, mysterious force. Nowadays it just means a fear so strong that you want to run away.

There are probably others, but these are the main ones.