Learn Spanish (Latin America): El cerebro es parte del cuerpo.

Even though you might think of location when talking about where the brain is, this sentence is defining what the brain is. For permanent characteristics, definitions, and classifying what something is, we always use ser (conjugated here as es).

Why isn't there an article before parte? Shouldn't it be una parte?

In Spanish, the phrase ser parte de (to be part of) usually drops the article. This is actually very similar to English, where we often say 'is part of the body' instead of 'is a part of the body'.

What exactly is del in this sentence?

del is a mandatory contraction. Whenever the preposition de (of) is immediately followed by the masculine singular article el (the), they must merge into del. So de + el cuerpo becomes del cuerpo.

Is the word parte masculine or feminine?

parte is a feminine noun (la parte). Because the article is dropped in the phrase es parte de, you don't see the feminine article here. But if you added an adjective, you would use feminine words, like una parte importante (an important part).