The Old School or OSR movement started around the time when 3rd Edition arrived and has continued strongly since. This has led to a lot of discussions and articles bout exactly what "old school" flavor is, and what (aside from the nostalgia factor and it being unique at the time) made 1st Edition so appealing. Whatever edition or offshoot you prefer to play, the fact that 1st Edition took root so strongly helped make it possible.
So what exactly is "Old School" tone? What makes a newer game "Old School"? I often see posters or companies barking up what I consider the wrong tree and calling it Old School, but the subject is very subjective to be sure.
I've opined at length over the differences between 1st and later editions, and what I like and dislike about the various editions. At times this discussion threatens to veer into some similar territory—so in those instances I'll try to keep to the point and not repeat myself.
In my opinion, "old school" is defined by a few principles:
1. DM caveat
The DM is boss. That doesn't mean unfair or boss-y, but the DM does call the shots. If a rule needs to be made, the DM lays it down with a minimum of page flipping. Player info is kept to a reasonable minimum. For example: players don't have magic item wish lists, instead they find unusual items, determine if they are magical, and figure out their properties.
Likewise, DMs are expected to carry their weight. Modules provide an adventure framework or skeleton and some details, but the DM must fill in the rest. The DM expects that the DMG will have a lot of good suggestions and examples, but that they will have to make up a lot of house rules to fill in the blanks.
2. Encounters need not be balanced.
Some encounters are easy. Some are not. Some require running. Players should have no expectation that all encounters are CR-approved. This doesn't mean the DM has license to kill off players willy-nilly—I've had passionate arguments with fellow DMs because of my dislike of that DM-vs-Player mentality that some DMs wear like a badge of honor (although I acknowledge that Gygax himself advocated that attitude at times). Playing "Old School" means is there is no safety net, and the players know this. If their DM is fair, they trust her not to drop them into no-win situations, but they also understand that dumb decisions—like not turning tail when faced with overwhelming odds—could mean destruction.
A good example of this came up in my 4e campaign, which I try to run with a more Old School feel. The PCs—3rd level at the time—were visited by a beholder. (Well, two actually.) Clearly presented with a much stronger enemy, the PCs stood down, except for a strong-willing minotaur that, well played, refused to lower his weapon. This meant consequences, yet it was not right for me as DM to punish the player for playing his PC true to form. And he didn't attack or do anything that stupidly put the party in peril. The player took just the right approach. So the beholder fired a warning shot, knocking the minotaur to the ground essentially unharmed except for his pride. (The minotaur then left his weapon on the ground, which was wise, as the next "disarming" would have probably taken the his arm off at the shoulder...)
3. Rules-light play
There isn't a rule for everything. As mentioned previously, the DM often must "wing it" and the players trust their DM to treat similar new situations in a consistent manner.
4. PCs can die. PCs can also be replaced.
Old School play can be far more dangerous (see #2 above). Save or die situations are expected. Living to reach 10th level isn't a given, but instead celebrated when it occurs. But if a should PC die, a new one can be created quickly. Which leads us to...
5. Fast character creation
PC options are limited and the character creation process for a 1st-level character should take less than 30 minutes and not require any specialized programs or spreadsheets. In fact, a calculator shouldn't be needed.
6. Hybrid characters and critters are rare
Multiclassing is a rarity and often not advantageous. Most PCs are single-classed. Likewise, there aren't many "leveled-up" or "classed" monsters, though they may have raised stats or use special items. Thinking on it, the Drow were one of the only races that seemed to display a lot of multi-classing in the old days. Pre-made multi-classed PCs did appear in many venerable modules, including the Tomb of Horrors, but they never came to the forefront (in 1e a 5th/5th magic-user/fighter facing off against either a 10th-level fighter or 10th-level mage could expect to be pulverized in a fair fight).
Likewise, players should embrace the personality of their character. In the old days, a player would hand a new character sheet to the Dm and say, "Here my new PC, Durlo" or perhaps "my new dwarf warrior Durlo" but now I often hear "I'm making a firedancer/ranger combo!" In latter editions, many players have come to think of their characters are bundles of skills & powers with a personality sheen, rather like bundles of cordwood tied with twine.
7. Randomness can be good.
Random tables for players and DMs ruled. Wandering monsters, baby! Everybody likes to roll dice and see the result on a chart, and the rules encouraged it. Having to create a high-level encounter on the fly because of a WM roll kept DMs sharp.
8. Narrative over checks
Want to search a room? Tell me exactly where you are looking, what you are touching (pin trap!), etc. None of this "take 10" or passive Perception check stuff. Old School is about a reliance on player description of actions and player-to-DM interaction, not skill checks.
9. Turn on the charm (and toss in some illusion too)
Charmed servitors and PCs enthralled by monsters was key. Charm spells could play a big role in encounters, as could illusion. An elf being charm-resistant actually meant something in the old days.
10. Off the grid
Forget that map of 5-foot squares. No sliding or pushing. Instead of moving with the precision of chess pieces, things are more loose and often subject to the DM's judgement. Scale is probably 10 feet to the square. A rough graph paper map (often wildly inaccurate) might serve for the session, and the minis may not be used every fight. On the good side, you'll be on the receiving end of far less of those pesky attacks of opportunity.
Of course, in the end, Old School is what you make of it. It's a style of play or adventure writing that brings to mind the elements that you wax nostalgic about and make you feel less jaded. And we could all use that from time to time.