Word for "a piece of a whole"

constituent (n)

one of the parts that form something

It is used as an adjective also:

serving to form, compose, or make up a unit or whole : component < constituent parts >

Its synonyms component and integral can be used also depending on the context.

Note: Integral is a bit tricky though. As a noun, it means the whole; but as an adjective, it has the connotation of the part of a whole. It is usually used with part.

of or belonging as an essential part of the whole; necessary to completeness; constituent: an integral part.

You also asked for "an object alone, but can be assembled into something greater."

Thus, I'm going to say synergistic element. Because synergy is:

the interaction of multiple elements in a system to produce an effect different from or greater than the sum of their individual effects.

Try Component or Components

As in "component parts"

I would call them building blocks or units.

a basic unit from which something is built up.