I won't go into the level of detail (names, actions) you're asking for, but this is how I see it:
In government, the Democratic establishment consists of those Democrats in the legislature who collectively choose the Majority/Minority leader in their chamber. This faction, collectively, wields their majority in the caucus to set the Democratic agenda, reward the productive, and punish the counter-productive. They decide which among their members get which committee seats, and which of the bills put forth get their support.
Outside of government, the Democratic National Committee obviously sets primary campaign rules, and probably does a lot of other less visible stuff in terms of raising money, setting agendas, picking winners and losers up- and down-ticket, etc. In addition, the DNC Superdelegates, as a class, can have a lot of influence over who gets what quid pro quo. You remember how Hillary Clinton made an effort to woo Superdelegates before the primaries had even started? That's because they're part of the Democratic establishment.
Finally, no discussion of an institutional establishment is complete without at least mentioning money. The biggest fundraisers, and the biggest donors, are going to have a lot of influence over the party. The owners and mouthpieces of the biggest media outlets on the left are probably also part of this category, but I'm not prepared to defend that bit.