Bernie Sanders is an independent, but he is able to seek the Democratic nomination and participate in the debates. Mike Bloomberg has said that he won't run for president because he is convinced that you cannot win the nomination without the support of one of the two major parties. Why can't he run as a Democrat, like Sanders? Please explain how you gain the privilege to seek the Democratic or Republican nomination.
Bernie Sanders is not running as a Democrat yet. He is currently trying to get the nomination to run as the candidate of the Democratic party, just like Hillary Clinton and Larry Lessig. Who will run for the party or not will be decided in the primary elections in 2016 which will be made among those voters who registered for the democratic party. Currently Hillary Clinton is leading the Democratic primary polls in most states, so it currently looks as if she will get the job, but we won't know for sure before the Democratic national convention which will take place from July 25th to 28th, 2016.
Also, Trump, Carson and Fiorina aren't officially endorsed yet. All are still competing for the nomination of the Republican party. Who will get it will also be decided in the primaries in 2016 among those voters who registered for the republican party. The party members can vote whoever they want, regardless of their official party affiliation.
But winning the primaries and becoming the officially endorsed candidate of a party is not the only way to run for president. One can also run as an independent candidate. But the chances of actually winning as an independent candidate are low. In the recent history of the United States no independent candidate ever got even close to getting elected. That's why Bloomberg says he won't run as an independent: he considers it pointless. And obviously he also considers his chances of getting the nomination by one of the two large parties out of his reach.
There's a few reasons the Democrats let Sanders run in their primary
- Sanders is essentially a Democrat in values. He caucuses with them in the Senate. A Sanders nomination would not have alienated core Democrats in the general election
Clinton was the well-established front-runner and there was a fear that she wouldn't have had any serious challenges prior to the nomination
Hillary Clinton, by contrast, was the presumptive Democratic Party nominee from the start. Bernie Sanders ran an impassioned and vigorous primary challenge against her. But her contest with Sanders was widely thought to be a warm-up, a sparring match, where Clinton could get her campaign going and the winning juices flowing.
Clinton had already sewn up all the core support from the DNC. While it didn't necessarily guarantee her a win, it definitely made the path far more difficult for Sanders
Donna Brazile, a former interim chairwoman of the party, says in a forthcoming book that an August 2015 agreement gave the Clinton campaign a measure of direct influence over the party’s finances and strategy, along with a say over staff decisions and consultation rights over issues like mailings, budgets and analytics.
The control was given in exchange for a joint fundraising pledge by the Clinton campaign that helped fund the DNC through the election year, Brazile says.
“This was not a criminal act, but as I saw it, it compromised the party’s integrity,” she wrote in a book scheduled for publication next week, a portion of which was excerpted Thursday in Politico.
A third party candidacy by Sanders would have been a disaster for the DNC, because he would have split the Clinton vote (the so-called Spoiler effect)
As to the other part of your question
Mike Bloomberg has said that he won't run for president because he is convinced that you cannot win the nomination without the support of one of the two major parties. Why can't he run as a Democrat, like Sanders?
Bloomberg was originally a Democrat who changed to Republican (he was running in the short time left on Rudy Giuliani's term after then Sept 11 attacks). He later changed to Independent. The problem is that voters often view repeated party switching dimly. Notable people who tried were Charlie Crist (ran for Florida governor 3 times in 3 separate parties, only winning as a Republican) and Arlen Specter (Senator, changed to Democrat in 2009 and then lost the subsequent Democrat primary despite DNC support). Bloomberg returning to Democrats for a presidential run would likely have been seen (and attacked) as opportunism.