A star is usually defined as a body whose core is hot enough and under
enough pressure to fuse light elements into heavier ones with a
significant release of energy.  The most basic (and easiest, in terms of
the temperatures and pressures required) type of fusion involve the
fusion of four hydrogen nuclei into one helium-4 nucleus, with a
corresponding release of energy (in the form of high-frequency photons).
This reaction powers the most stable and long-lived class of stars, the
main sequence stars (like our Sun and nearly all of the stars in the
Sun's immediate vicinity).

Below certain threshold temperatures and pressures, the fusion reaction
is not self-sustaining and no longer provides a sufficient release of
energy to call said object a star.  Theoretical calculations indicate
(and direct observations corroborate) that this limit lies somewhere
around 0.08 solar masses; a near-star below this limit is called a brown
dwarf.

By contrast, Jupiter, the largest planet in our solar system, is only
0.001 masses solar.  This makes the smallest possible stars roughly 80
times more massive than Jupiter; that is, Jupiter would need something
like 80 times more mass to become even one of the smallest and feeblest
red dwarfs.  Since there is nothing approaching 79 Jupiter masses of
hydrogen floating around anywhere in the solar system where it could be
added to Jupiter, there is no feasible way that Jupiter could become a
star.