The Forgotten Realms is the current "default" setting for Dungeons and Dragons in the current 5th edition. It was also the default setting in 2nd edition. In many respects, it's the most popular setting for D&D - it has more sourcebooks and more novels written about it than any of the other settings.
If you're new to D&D, or new to the Forgotten Realms (FR) then here are some places to go for information. Most of the books, adventures, and sourcebooks focus on the northern end of the continent of Faerün.
If you're looking for more information about the Forgotten Realms, the most popular setting for Dungeons and Dragons, the most comprehensive repository of information is the FR Wiki. The folks there have organized EVERYTHING from the sources below and many other sources. The Wiki is great for looking things up, but it's not exactly easy to just start reading it.
This is an excellent companion page to the wiki. It's taken much of the wiki's geographic resources and put them on a map.
Wizards of the coast has also released a free map of the realms. It's an excellent resource to learn where many things in "The North" are located. It's not the whole area of the Forgotten Realms though.
Additionally the cartographer Mike Schley has many FR maps that he's drawn for official products for sale.
Official 5E Sourcebook
For 5E, the Sword Coast Adventurer's Guide has good updated current-year* information for the northwest side of the Forgotten Realms. Plus it has good overview information about the realms that will give anyone a good grounding. Additionally it has 11 subclasses, 12 backgrounds, 9 races/subraces, and 4 new spells that are fine 5E materials.
Drive Thru RPG has nearly everything from previous editions for sale as PDFs.
Forgotten Realms Campaign Set (1e)
Player's Guide to the Forgotten Realms Campaign (2e)
Forgotten Realms Campaign Setting (3e)
This is (IMO) the best option for a good overview of the FR BUT time has passed and some things have changed.
Forgotten Realms Player's Guide (4e)
In 4E there was a cataclysm in the FR called the "Spellplague". This caused a lot of changes in the FR, many of which were undone in 5E.
Additionally, there's about a hundred Forgotten Realms novels. Ed Greenwood and R.A. Salvatore are some of the more famous authors, along with Jean Rabe, James Wyatt, Bill Slavicsek, Bruce R. Cordell, Drew Karpyshyn, Richard Baker, Elaine Cunningham, Erik Scott de Bie, and Kate Novak.
You can find out more about the novels here:
* Current Year in the Forgotten Realms
We're not entirely sure what year it is RIGHT NOW, but Princes of the Apocalypse is set in 1491 DR (Dale Reckoning, one of the more popular calendars in the Forgotten Realms).
My personal headcanon is that the major adventures each happen once a year, but the anthologies don't count because they could take place at any time.
Out of the Abyss 1492 DR
Curse of Strahd (doesn't get a year because it takes place in Ravenloft)
Storm King's Thunder 1493 DR
Tales from the Yawning Portal (doesn't get a year because it's an anthology)
Tomb of Annihilation 1494 DR
Waterdeep: Dragon Heist/Dungeon of the Mad Mage take place the same year, 1495 DR
Ghosts of Saltmarsh (doesn't get a year because it's an anthology)
Baldur’s Gate: Descent into Avernus 1496 DR
EDIT: Chris Perkins, who is just about THE top banana for Dungeons and Dragons apparently said that Dragon Heist takes place in 1492. ¯\_(ツ)_/¯ Apparently things are happening fast in the Forgotten Realms.