Gaming

D&D fifth edition rules

This is the fifth version of D&D, released a few years ago I believe. Each version of D&D is called an "edition" and each one contains changes & new rules, characters, settings, stories, etc. Think of it like an update to a video game. Some people prefer old editions, some like new editions. The rules in BG3 are mostly from 5th edition (abbreviated as 5e). Like with video games, the publishers of D&D are called "Wizards of the Coast" so when people refer to editions, they refer to updates released by that particular company. Other companies make other versions, modifications, and campaigns within and like D&D, but only WotC makes D&D editions.

Forgotten realms

This is just the setting for D&D. It's rather high fantasy, and if you're playing a bog standard D&D game in real life, this is probably where your story is going to be set. Most of the settings within the Forgotten Realms are set within the large continent of Faerun. FWIW, "Baldur's Gate" Is the name of a canonical city in Faerun. It's a very wealthy and prosperous merchant city state. There are other campaigns and stories from other continents in the Forgotten Realms (and from beyond the forgotten realms), but Faerun is by far the most fleshed out.

TL;DR: 5e is the "fifth edition," which is the most current "official" ruleset for the game. The Forgotten Realms are the official setting for the game. Faerun is the main continent, and Baldur's Gate is a city on that continent.