Most of the reports I've seen have stated that the decision gives corporations, probably including non-profits, and possibly unions the right to speak during the period immediately prior to an election.
The primary Act that was overturned in effect prohibited anyone not a party to the candidates' campaigns from any public statements during a period, generally 90 days, immediately preceding any election that qualified under the provisions of the Act.
One effect of this was that a candidate could "smear" an opponent on issues not central to the opponent's platform but of major concern to "issue groups," with the expectation that the oponent's campagin would be "too busy to respond," and those with concerns were prohibited from "rebutting the lies."
Having downloaded the full text of the decision, and after spending about 3 hours thus far attempting to read it, I can't quite be sure what it does accomplish. The decision is 185 pages in the PDF and is not exactly an example of clear exposition.
It does appear, however, that the majority opinion examined and overturned some two dozen prior SCOTUS decisions contrary to the decision that the conservative judges wanted. The majority significantly extended the scope of a handful of prior precedents that supported their position.
In a significant departure from prior Court decisions, SCOTUS appears to have rendered decisions on issues not part of the appeal petition, thereby assuming an "activist court" stance that has been a core fear and complaint of conservatives - to act clearly in favor of conservative agendas not legitimately part of the questions before the court.
It has been a "settled precedent" that as an appeal court, SCOTUS cannot address a question that is not directly included in the petitioners' appeal petition. That precedent apparently has also been overturned.
It is NOT CLEAR that the decision gives any rights to labor unions. In a couple of places, that I found so far, it could be construed that the majority opinion intentionally was vague, or implied contrary intent, on this point. PACs (read lobbyists) are mostly incorporated as non-profit corporations, and while it seems that the decision intends to free them from all restraints, I'll have to do quite a lot more reading to be sure.
As the decision is very densely written (you guess whether that's a pun?) it will take me some significant additional time and effort just to complete reading it. Tracking the citations and looking at the precedents that were discarded quite probably will be more than I can expect to accomplish. One might wonder if that was intentional.