A vaishnava is someone who worships Vishnu, thats the meaning of the word. Other meanings are someone who belongs to, devotes himself to or belongs to Vishnu.
To understand who or what Vishnu is, you have to understand the vedic system of the gods, in the vedic system there are many gods and Vishnu is one of the three main gods, which are Brahma, Vishnu and Shiva. Those gods form a trinity, where each god has a different task in the creation and represents a different guna or mood. Brahma is the creator and represents the guna of passion, Vishnu is the keeper and represents the guna of virtue/knowledge and Shiva is the destroyer and he represents the guna of ignorance/darkness.
Those three gods were created by Krishna the highest god to create the material universe, where Brahma creates the material universe and Shiva destroys it, where as Vishnu is neutral and preserves it. Vishnu has kind of a transcendental position since he is an incarnation of Krishna the highest god, or Krishna is an incarnation of Vishnu, but it does not really matter since he is transcendental anyway.
If you can use your common sense what principle you want to worship and what you want to become, it should be a no-brainer what to chose from that three principles and which god to worship, which is of course Vishnu who represents knowledge and virtue. An intelligent person would always chose virtue and knowledge. Only the mood of virtue leads to true happiness, the mood of passion(Brahma) leads to some short term happiness mixed with suffering and the mood of ignorance (Shiva) leads to suffering all the time.
Ironically vaishnavas are probably the rarest kind of devotees and most people worship Shiva, which is the most popular god amongst materialistic/egoistic people, since he gives blessings easily, while Vishnu is not popular, since Vishnu will not help you with material progressions, since he knows that it is bad for you.
So if you want to be a good person, you worship Vishnu and if you want to be a bad person, you worship Shiva, it is that simple. ( Well in reality the universe is a bit more complicated, but this assumption is a good place to start )