Over at /r/baldursgate they discuss the original games that triggered all this nostalgia: Baldur's Gate 1 & 2, Icewind Dale 1 & 2, Planescape Torment. Importantly, all those games (except Torment, but read on) have been revised in the last year by Beamdog Games, so that they will run on modern laptops & tablets with minimum fuss. I believe they are on Steam, though Beamdog may still want you to buy them from them directly (more $$$ goes straight to them).
Baldur's Gate
The Baldur's Gate games are party based (assemble a team, each member will have his/her own goals & personality). These games are a balance of role play & cool combat encounters. The role play is serious, too. It might feel sparse compared to some modern games (there are times when you are wandering the wilderness talking to no one for a while), but the game will impose real consequences. For example, hire on Viconia and a paladin (Anomen or that older guy) and they will literally try to murder each other, and you will have a smaller team because of it. You can't put anyone together -- they act like real people and will conflict if you try to get polar opposites to get along. Also, if you jack up your charisma and are basically gorgeous and well-spoken, they might stick around longer because frankly it's true that hot charismatic people can get people to follow them much more easily & often.
Another example from BG 2: if you say the wrong thing to one particular person, the module will skip you over an entire section of the game. It just is closed to you as an option (the island? I think...) that you cannot get back. So what you say to people matters, and will change the game.
Another example from BG 1: If you keep in your party the evil Monty and ...whoever that other bad guy is... then when you have the conflict with the Red Wizards, they will get you out of it. Very few people know this, because Monty & his friend suck, so nobody keeps them around. But there you go: consequences for your choices, branching gameplay based upon what you do.
In BG 1, archers win at everything. In BG 2, wizards win at everything.
Icewind Dale
The Icewind Dale games are different. There are no NPCs to recruit into your team. If you want a team, you need to build the character sheets yourself. Because of this, you can expect no intra-party banter, no clashes of personality, and no side-quests based upon party members. These games are not a good balance of role play and combat, but that doesn't mean they suck. They are fun. They are very fun actually, but they focus on combat and puzzles. Very little role play. You derive enjoyment from making good characters who are optimized for skirmishes and battlefield control. You explore interesting locations, and kill everyone there. The music is great.
In IWD 1, clerics win at everything. In IWD 2, I'm not sure what wins at everything.
Planescape Torment --> Torment: Tides of Numenera
This is the only game that Beamdog isn't renovating, as far as I know. However, we don't care because of /r/numenera -- that is, there is a new Planescape Torment successor coming out called Torment: Tides of Numenera. This game is very much like Pillars of Eternity, right down to lots of the same people working on it. It was also funded by Kickstarter (in fact, OP notes in his original post that Pillars is the 2nd most funded game on Kickstarter... the first is Numenera). Numenera is different from Pillars in one big way: it has a role playing game behind it already. In other words, Pillars is made up from whole cloth, fully original, but Numenera is based upon the tabletop role playing game called... Numenera. So while Tides of Numenera won't be out for a few more months, you can technically play it right now by grabbing the manual, getting some friends, making character sheets with pencils & paper & dice, and start playing.
Chris Avellone, who is working on all these projects, described the difference like this: "Numenera is much more free flowing, much more story-focused, and Eternity is stuff like dungeon exploration, party team, how do you approach a problem, how do you approach an encounter." So basically, Numenera is your BG series, and Pillars is your IWD series. That's an oversimplification, but if you wanted a broad overview, that's the easiest way to boil it down to a sentence.
Good luck with all the games! Have fun!