Open stage night is a term being more commonly used for nights which have been derived from other similar evenings like, open mics

Open mike

An open mike or open mic is a live show where audience members may perform at the microphone. Usually, the performers sign up in advance for a time slot with the host or master of ceremonies. These events can be focused on poetry and the spoken word, music, comedy, and the open format of open...

, acoustic nights and poetry slam. The main difference being that usually if the term open stage is used then the night is open to a wider platform of performers, thus the word "open". So at an open stage you can find anything from dancers to poets, singer songwriters to puppeteers, all kinds of performance from anyone with the inclination to perform are usually welcomed.
Similar to its sibling nights, this kind of night usually also has a presenter

Presenter

A presenter, or host , is a person or organization responsible for running an event. A museum or university, for example, may be the presenter or host of an exhibit. Likewise, a master of ceremonies is a person that hosts or presents a show...

who allocates time slots for performers and who among other things can be seen, judging the mood of the night, entertaining, and introducing acts.
Some open stages like to embrace the idea of also the position of presenting being open also, so that if somebody expresses a desire before the night to be presenter, then they should be just as welcome to have a go taking that position also.
Some open stages also like to consider the evenings free, as it cannot be truly open if somebody financially hindered can not enter.