First things first, WotC has some weird editions nomenclature that fails to reflect the development of the game and really should be just ignored. As far as I can tell, it's something like:

Quote Originally Posted by WotC
"1st edition" - Everything up to and including 1st edition AD&D.
"2nd edition" - The 2nd edition of AD&D.
"4rd edition" - This is the first one we've done ourselves.
... or something of the sort. Just ignore that, really. Instead, refer to this Wikipedia page.

Quote Originally Posted by BIGMamaSloth View Post
1. Are AD&D and 2nd edition the same thing? What are the core book(s) to play called?
AD&D has two editions: 1st and 2nd (obviously, the 1st one didn't actually call it that, since they didn't know there will be a new one later on. It was just called "AD&D".)

If you compare the core books of AD&D 1st and 2nd, they're very, VERY similar with a few differences. In 1st ed., the primary means of resolving attack rolls is to look up the result on a chart, and THAC0 is only offered as an optional shortcut to that; in 2nd, THAC0 is the assumed resolution mechanic. In 1st ed., Thieves have a fixed rate of improvement for their skills, in 2nd, you can freely distribute so-and-so-many points at every level. In 1st, Illusionists are a separate class with a separate spell list, in 2nd, specialised Wizards and Clerics work somewhat differently. Monks and Assassins have been cut from 2nd edition. 2nd ed. has Non-weapon proficiencies for a skill system, 1st ed. doesn't. There are other differences of the sort, but by and large the two editions are interchangeable with minimal effort.

If you include the later non-core additions to 2nd edition, the difference starts to grow, mainly by the introduction of kits (which modify your class), playable humanoids, etc.. The underlying mechanics are still pretty much the same, but backporting a 2nd ed. character into 1st might include chopping off a bunch of bells and whistles.

Another difference was in design sensibilities, which is mainly obvious in adventure modules and 2nd ed.-era novels. I don't want to go into it in too much detail, but long story short, 2nd ed. was reacting to the anti-D&D witchhunt of the 80s and playing it safe and Politically Correct.

Core books... For both, Player's Handbook, Dungeon Master's Guide, Monster Manual. For 1st edition, Unearthed Arcana, Fiend Folio, Monster Manual II and Deities & Demigods also saw heavy use, but are not strictly core.

2. Are OD&D and 1st the same thing? What are the core book(s) to play called?
Nope. Original D&D is the very first one, the 1974 game on the Wikipedia chart.

There weren't all that many books for OD&D in the first place, so the notion of "Core" doesn't really apply. There's Dungeons & Dragons Volume I: Men & Magic, which is the fundamentals. You also want to have a copy of Chainmail (say, 3rd edition), the wargame on which D&D is based - Volume I makes references the Chainmail rules without actually including them. There were also 4 supplements (Blackmoor, Greyhawk, Eldritch Wizardry, Gods, Demigods & Heroes); they're not strictly necessary, but if you want to have a print copy of OD&D, you probably also want to try and get these. Also, Swords & Spells are miniatures rules, but I really don't know anything about other than its existence.

3. which one of these two had non-human races as classes? both?
That would be OD&D and Classic/Basic/BECMI/Rules Cyclopedia (nomenclature varies a bit, it's the one on the right side of the Wiki table between '77 and '95.

4. Are 1st and second edition very dramatically different? like 3rd and 4th were?
If you mean the 1st and 2nd editions of AD&D, already addressed that. Not very dramatically, overall, and if you don't use 2nd ed.'s splatbooks, very little, actually.

5. Can anyone give me a general feel or idea about these systems to help me decide if I'd like them?
OD&D: Very basic. You could have a look at Swords & Wizardry, a free retroclone which is pretty much the same thing. Personally, I appreciate OD&D / S&W because it's so pared down, I can easily tinker with and add things to it.

Basic: This is really its own beast, and from what I can tell, something of the red-headed stepchild. This is the only edition of D&D (excluding some modern retroclones) which genuinely tried to do something new and interesting about high-level characters. The Mentzer series (than the Rules Cyclopedia) had all sorts rules for running a PC's domain, conducting large-scale battles, questing for immortality and the like. In fact, the "I" in "BECMI" refers to Immortals, a book solely dedicated to playing characters who have already achieved godhood and can now travel between exotic planes of existence, mix it up with other gods, create their own servant species, create bodies like they were clothes, etc.. It almost its own little separate game ruleswise, and it explores an area other editions of D&D have left alone.

AD&D: This is the common language an entire generation speaks, and the edition that really became a smash hit. It offers more mechanical options than OD&D or Classic, but still isn't nearly as fiddly a WotC's versions. Note that in actual play, many options like Weapon Speed, Weapon vs. Armour Type, Weapon size, etc.. are ignored. All in all, it (like all other TSR editions) run combat much, much faster than WotC's D&D. In my personal opinion, it's also the first one to really offer some implicit, covert suggestions on a certain style and sensibility for the game, a mixture of various literary influences.

AD&D 2nd edition: If you use kits and the Player's Option rules, it's a bit closer to 3E insofar that creating your character becomes somewhat fiddlier. The "standard" feeling for 2nd ed. games is, in my opinion, blander and less quirky than in AD&D 1E, thanks to the "D&D is Satanic!" scare. What I find of the greatest value in this edition is the large variety of settings that came out for it - Forgotten Realms is kind of meh ("Let's have a Greyhawk but minus everything that might offend soccer moms!"), but you also have Dark Sun, Planescape, Speljammer, Dragonlance (apparently it's not really bad if you ignore the novels and the untolerably railroady official adventure modules), Masque of the Red Death (gothic Earth), various exotic parts of the Forgotten realms (Mesoamerican Maztica, oriental areas, Arabic Al-Qadim, Spanish California + mutants Red Steel, etc..) Not all of these are of equal quality, but they're certainly good for cherry-picking.