Every format has a unique feel, and the fact that there are so many interesting ones is definitely one of the things that makes Magic great. There are two well-known eternal formats (Legacy and Modern), an emerging new one (Frontier), and one that allows the power nine, expensive cards from 1993 (Vintage).
One good way to compare them is to understand that Magic has gone through several eras of card design philosophy. Each era had a different balance between creatures, non-creature spells, and mana. Additionally, the blue spells (which can kind of manipulate the game) were different in each era.
One blue spell archetype is counterspells: cards that cancel the effects of a spell cast by the opponent. Each set has two or three versions of this; some cost very little mana and some have very few drawbacks. Since Legacy players have access to all of these, there are several incredibly good ones that are utilized, and players will often get into exchanges where they counter each other’s counterspells to try to get their original spell to resolve. To do this effectively, a player can guess which counterspells their opponent probably has, monitor their behavior to determine which might be in their hand, and then try to pick a moment when they either can’t or won’t want to deploy them. For this reason, Legacy is sometimes compared to fencing — where a flick of the wrist at just the right time leads to a touché.
There are also non-blue decks that regularly show up in Legacy but they struggle to best the blue-based ones; a current blue favorite is “Omni-Show”, where you use a card that lets each player cast something for free, to play a card that lets you cast everything for free. But despite being able to cast anything, the deck uses just one card at that point: a 15/15, un-counterable titan that gives you an extra turn. The rest of the deck is just cards that help you find these three. I’m sure this can be a fun puzzle to solve, but in the more linear, larger context of Magic it’s a bit bizarre.
There no mistaking the beauty of early Magic art, and the design of the first cards is just as splendid to the senses, which has led to immense nostalgic value in any vintage format, including Old School which uses only cards from '93 and '94 - the original set and first four expansions. The cards from then were shrouded in a little more mystery, and the mid-nineties were a time when we were discovering this completely new way to play, imagine and construct games. The game has gotten better, but it was already really, really good to begin with, and the original editions have truly stood the test of time, to the point that they can sensibly be considered the purest way to play. Not everyone would feel that way of course, but at least it's in the discussion. It's also in the discussion for most powerful accelerants and spells, so for people who simply like to have all the best toys, well Vintage decks are one of them. They frequently cost more than a car. But unlike cars, they never decrease in value.
Magic Online is probably the most common place that Legacy is played these days, although in-person play is still available around the world and is even returning to the Pro Tour.
Modern draws from the pool of cards that have been printed since Mirrodin was released in 2003. By then, Wizards had stopped printing blue spells that were insanely powerful relative to the rest of the field. But, because mana costs are in whole numbers, either 0, 1, 2, or 3, etc but not 1.2 or 2.5, there are a few spells in each set that are hyper-efficient, and when you give players access to several dozen sets, there are busted things the decks can do. Modern’s meta-game has many archetypes, but many of them are designed to rapidly eke out a turn-4 win; competition is fierce.
One deck that has been dominant is “Infect”, which utilizes creatures that deal damage in the form of poison counters. Poison counters, despite originating in Magic’s first sets, are usually a fringe mechanic: if a player gets 10, game over. The infect mechanic, by enabling creatures to deliver multiple poison counters at once, arguably changes the opponent’s starting tally from 20 to 10; this advantage is an unmistakable part of the format that not everyone likes.
Many of Magic’s balances are based around having a little time to recover from an early onslaught. That means you can lose 10 or 15 life and still be fine. By changing that equation, Infect has hurt a lot of feelings! Many consider it to have been a mistake. And yes, at times when the design team has made mistakes that annoyed the community, they have corrected them by banning problem cards. More recently, bannings have been used to balance out decks that get too powerful (ahem, blue spells). On the other hand, there are dozens of different decks that have a fighting chance against each other. In fact, in the time since I began writing this Modern has gone from nearly being a hard pass to apparently having a banner year.
A lot of people enjoy competing in Modern events. But with so many cards in the pool, there are bound to be degenerate (difficult-to-interact-with) combos, everywhere. I’m not fully convinced that addressing them is a puzzle that I can recommend. However, navigating unforgiving matches can certainly improve one’s play. I’ve definitely seen some indications that after a few years of experience, Modern is something that starts to appeal to players because it allows them to have such a high-powered, fast-paced exchange to test themselves. And the same thing is almost certainly true of Legacy (as well as Vintage). Without a doubt, when you see how much money people sometimes spend on their competitive decks, you can be pretty confident that there are some amazingly fun reasons they’re doing that.
Frontier was recently created by a Japanese card store, because cards for Modern are hard to acquire in Japan. Frontier allows cards from expansion sets starting with 2014’s Khans of Tarkir and Magic Origins. Therefore, the card pool is not yet hugely different from Standard, but as time passes it will diverge. And, there is one huge difference: the legality of fetch lands.
Fetch lands are one of the most elegant solutions to mana-fixing that Wizards has ever concocted. They allow you to search for one of two land types, and play the land untapped (so it can be used immediately), at the cost of 1 life point. By doing this, they also “thin your deck”, so there’s one less land that you will draw later in the game; and gather cards in the graveyard, which is a technique that can pay off in some decks. Ostensibly, you can fetch basic lands, which produce only one color of mana. However, a few times in the game’s history, nonbasic dual lands have been printed that carry the designations of the 5 basic land types. Therefore, if I have a fetch land that allows me to fetch a Forest or a Plains (Windswept Heath), I could choose to fetch a “Forest Mountain” (Cinder Glade) or a “Plains Island” (Irrigated Farmland). With two such land cycles legal in the format, each fetch land gives the player access to any of four colors of mana — and these lands can often enter the battlefield untapped, enabling an uninterrupted start to the game (actually, uninterrupted isn't a perfect word to use here, because the extra shuffling time resulting from fetching lands has slowed down tournaments enough to cast doubts on whether they'll ever return to Standard).
Deck construction in Frontier, therefore, isn’t really bound by the normal burden of mana-fixing that restricts most other decks to two or three colors. Four- and five-color decks are also competitive in this format. Your deck design is only limited by your imagination.
Frontier hasn’t taken off yet, so it remains to be seen whether it will be possible to compete at big tournaments (or even local ones) with these decks. However, my guess is that it will become popular, and eventually may play the role once held by Legacy and Modern as a premier competitive format.