Historically presidential elections haven't had the same quick communications that exist now. Elections tended to work by picking delegates who would travel to conventions and negotiate their way to a common candidate. This resolved the problem with split fields. Note that most democracies pick prime ministers by a variant of this method. The US is different in that the delegates only pick the presidential candidates and the legislators are picked separately. In parliaments, the delegates and legislators are the same.
In the Democratic primaries, any candidate who gets at least 15% of the vote gets a proportional share of the delegates. The delegates per state is determined by a formula that the party uses. There are also a number of superdelegates who were elected to some other position and given a vote at the convention. The superdelegates do not have any special powers, just the same vote as other delegates. They are super in that they don't have to run separately to get appointed as delegates. You will sometimes see them referred to as unpledged delegates, meaning that they can change their vote at any time.
The delegates choose a nominee by repeating the vote until someone gets a majority. In the first ballot, delegates are generally bound to the candidate to whom they are allocated. Most delegates would be freed in later ballots. Every convention has chosen its nominee on the first ballot since 1952.
Sanders needed to get 60% of the delegates in the remaining primaries before Tuesday. That's going to be higher now.
There are 4051 delegates in the Democratic primaries. 1979 delegates have been allocated already, leaving 2072. Sanders needs 2383 delegates to win and has 828 so far. This leaves 1555. So Sanders needs to win roughly 75% of the remaining delegates. Of course, this assumes he gets 0 superdelegates. Half the 714 superdelegates would be 357, so he'd only need 1198 more delegates. That's only 58% of the remaining delegates. Of course, most indications are that the superdelegates prefer Clinton.
To reiterate, to be safe, Sanders needs 75% of the remaining delegates to be sure. To be competitive, he needs at least 58%, but that puts a lot of reliance on his ability to convert superdelegates. My personal feeling is that he would need 65-70% of the remaining delegates and roughly a third of the superdelegates.
The truth is that Sanders did not do that well in Michigan, except in comparison to expectations. He needed something like 60% of the vote and got a bit over 50%. Since he was trounced in Mississippi the same day, he actually finished the day further away from the nomination. The only reason that they described Michigan as a success was that Sanders had been expected to lose by about 20%.
Media coverage makes it look like Sanders is closer than he really is. Its concentration on winning states like they were sports matches overestimates the impact. He did not do as well in Michigan as he needed to do. On March 15th, he did even worse, not even meeting the original delegate targets outside Missouri — much less compensating for his previous shortfalls.
While it is still possible for Sanders to win, it is quite unlikely. He'd have to win every state like he won Vermont and New Hampshire, which were demographically two of his three best states.