Vaishnavism refers to worship of Vishnu. A devotee of Vishnu is a Vaishnava.
In general, Vaishnavas regard Vishnu to be the absolute reality (brahman) who incarnates as various avatars. The avatars appear in the world when there is decay in morality and righteousness in the world. The most well-known avatars of Vishnu are Rama and Krishna.
There are various vaishnava sects and belief systems. Srivaishnavas, for example, align themselves with vishisht-advaita (qualified non-dualism) of Ramanuja – which posits that while the world is illusory (maya), it is also the means by which to connect with Vishnu – the absolute reality.
Followers of Dvaita Vedanta (founded by Madhavacharya), in contrast, believe there are two separate realities – Vishnu who is the supreme self and a second reality of the individual soul and matter, that is dependent of the first. To this sect, moksha (liberation) is the realization that the finite world is ultimately reliant on the supreme (Vishnu).