In Mandarin, the word for word is 词 (Cí). It's pronounced somewhat like "tser". An example of its use is in the word 词典 (Cídiǎn) which is the word for a dictionary.
Individual characters are called 字 (Zì) but that can also be used for a word. Mandarin characters would be 汉字 (Hànzì) where 汉 (Hàn) represents the Han people ie: most of the Chinese.
Most of the parts of English speech have fairly direct equivalents. A Noun is called 名词 (Míngcí) which pretty much means "name word".
A verb is called 动词 (Dòngcí) which is close to meaning a "move word" or perhaps a "movement word".
An adjective is called 形容词 (Xíngróngcí) which is a little more complex but basically a "describe word". 形 on its own is a shape.
An adverb is called 副词 (Fùcí) which is close to a "assistant word", a "secondary word", or an "auxiliary word". 副 on its own is means secondary, auxiliary, deputy, assistant, etc.
Learning Mandarin
I'll write more soon on the best methods for learning. If you want to get a taste for it in the meantime though, my current favorite is Tutor Ming. If you decide to try it, click here and it's a bit cheaper for you, and for me.