A main sequence star is a star in the phase of its life termed the main sequence (MS), which is when it is burning hydrogen into helium through the proton-proton chain or the CNO cycle at its center, its stellar core. Such stars fit on a diagonal region (roughly a line) on an H-R diagram, the region/line on the diagram also being referred to as the main sequence. For many stars, this is the majority of their fusion lifetime, e.g., billions of years for Sun-like stars.
The term main sequence star is also likely occasionally used in a general way for a star that has or will have a main sequence in its lifetime, even if it is before or after this phase, to distinguish it from, for example, brown dwarfs. The unqualified term dwarf star actually refers to any star that is not a giant star, generally a star in its main sequence. If qualified, it depends: a white dwarf is beyond its main sequence, a brown dwarf has no main sequence, but a red dwarf is indeed in its main sequence. The suffix V attached to the spectral type indicates a main sequence star, aka a dwarf star. (Thus, do not take V as an indication of a variable star: rather it is a luminosity class.) For example, the Sun's spectral type has been cited as G2V.
Before their main sequence, such stars are powered by gravitational collapse and termed pre-main-sequence stars. The time-length of a star's main sequence varies based upon the mass of the star, more massive (early) stars producing much more fusion, more than making up for the larger amount of hydrogen, resulting in a shorter main sequence. The resulting main sequence lifetimes vary from millions of years to hundreds of billions. After the main sequence, all but the smallest stars (generally, all larger than a red dwarf) proceed to giant-star phases (post-main-sequence stars), while it is presumed that red dwarfs, none of which are old enough to have left their extremely long main sequence, will simply cool down.