Beshert means "inevitable" or "preordained." It can apply to any happening which appears to bear the fingerprints of divine providence, such as bumping into an old friend you were just thinking about.
But it is used most commonly about marriage and shidduchim ("matches"). Singles pray to "meet their beshert," their life partner, the other half of the broken eggshell with whom they will find love and fulfilment.
The fatalism implied by this image doesn't sit well with Judaism's emphasis on free will and responsibility for actions. Kabbalistic sources modify the simple picture, stating that we may have more than one beshert, and that we can lose or gain a beshert by our good and bad actions and prayers. We can fail to recognise him or her, too.
But Jews do recognise the need for, and availability of, divine help in bringing us together with our marriage partners. The Talmud says, with maybe a touch of hyperbole, that since the work of Creation was finished, God has been engaged in matchmaking, and that the task of bringing together a couple is tougher than parting the Red Sea (Sotah 2a).
Beshert is a Yiddish word whose etymology is not entirely clear. The most likely source is the German "beschert," which means "given." "Beschert" is often used to mean Christmas and New Year presents, which according to folklore are divine gifts, hence the connection to beshert.