This question has two different answers so we will try and explore both of them. First of all, for an object to qualify as a star it has to be a ball of gas large enough that fusion is triggered at its core. We think for this to happen the object would need to have at least 80 times more mass than the planet Jupiter. This fits with what we have seen out in our universe. Currently the smallest star we have found is called OGLE-TR-122b. This star is around 96 times more massive than Jupiter. We call this type of star a red dwarf.
The other side to this answer involves a star that is at the very end of its life. The type of star we will discuss now was once one of the biggest stars we know of. When a very large star runs out of fuel, the core stops reacting and the force holding the star up ceases, causing it to collapse. As all this material comes crashing down, the core of the star gets compressed. If the star is large enough then the core gets compressed until even the atoms inside are squashed together. This leaves a huge ball of neutrons. This material forms some of the densest material we know of. These compressed cores can contain as much as twice the material in our sun, squashed down to less than 20 kilometres (12.4 miles) across.
Answered by Josh Barker from the National Space Centre
Image courtesy of NASA