The subject is the main actor in the minor drama that is a sentence.
It is the main noun (person,place or thing) in a sentence, and it usually initiates the action taking place in the sentence.
If a sentence were a car engine, the subject would be roughly equivalent to the gas in the engine. It is what makes the sentence go. If a sentence is broken, just like a car engine, the first thing you want to check is the gas [the subject].
The subject can come in many varieties.
It can be a single, stand-alone noun:
John Fitch ate his hat for lunch.
It can be a compound noun joined by a conjunction:
The toad and the painted turtle jumped off the log together when Louise paddled by in the canoe.
It can be a gerund phrase:
Falling out of the sky is fun but only the first time you do it.
It can be an infinitive phrase:
To err is human.
It can be a prepositional phrase:
Over the rainbow is where I want to be.
Remember: although one can have subjects within many types of clauses [dependent clauses, relative clauses, noun clauses], there is only one main subject in a sentence.