When describing spatial relationships, the words 之间 (zhījiān) and 中间 (zhōngjiān) will come in handy to express the concepts of "between" and "among." Both have a few particular uses, though.

Both can mean "in the middle"

Grammatically, both 之间 and 中间 are nouns (location nouns, to be precise). One difference between the two, however, is that 中间 can be used all by itself, but 之间 is always used together with other words.
Between two points in space
之间 and 中间 can both refer to a space "in the middle" between two points, as long as we're talking about physical space. So between Shanghai and Beijing, between my eyes, between the psychotic monkey and the enraged elephant, etc.
Structure

Examples

Zài shā fā hé zhuō zi zhī jiān yǒu yì zhī māo。
在沙发和桌子之间有一只猫。
There's a cat between the sofa and the table.

For time and number ranges use 之间
This usage refers to examples like "between 8am and 3pm" or "a number between 32 and 51."
Structure

Examples

Nǐ xū yào zài bā diǎn dào shí diǎn zhī jiān lái.
你 需要 在 八 点 到 十 点 之间 来 。
You need to come between 8:00 and 10:00.

For describing location within an object, use 中间
This "within" could be "inside" a space (like inside a house, or inside a refrigerator), and is similar to 里面 except that it is also centered in the space (with regards to the sides, but not height). But it can also mean "in the middle of" a two-dimensional physical space, like in the middle (center) of a table, or in the middle of the screen.
You can't use 之间 in this way.
Structure

Examples

Fánɡ zi de zhōnɡ jiān yǒu yī ɡè dà zhuō zi .
房子 的 中间 有 一 个 大 桌子 。
In the middle of the apartment there's a table.

Other uses of 之间

之间 can be used to indicate relationships
The idea is that the relationship is "between two people," and the "between" is indicated by 之间. This usage of 之间 can also refer to relationships between a person and a company, two companies, two countries, etc.
You can't use 中间 in this way.
Structure

In the structure above, Noun1 and Noun2 are frequently people.
Examples

Wǒ men zhī jiān shén me huà dōu shuō.
我们 之间 什么 话 都 说 。
The two of us can talk about anything.

Other uses of 中间

中间 can also mean "among"
中间 is used when referring to one (or more) "among many."
Structure

Examples

Nǐ men zhōnɡ jiān yǒu huì chànɡ ɡē de mɑ ?
你们 中间 有 会 唱歌 的 吗 ?
Can anyone among you sing?

中间 can be used as part of other words
中间 is a noun, and can combine with other nouns to create a number of fairly common verbs.
Structure

Examples

zhōnɡ jiān rén
中间人
intermediary (lit. "middle person")