I have been playing Standard Constructed (Type 2) games so far but it seems that there are a lot of formats which are gaining popularity. I'm looking for the most popular ones which are recognized officially as well so I can attend competitive matches.
WotC has a page with basic explanations of the different sanctioned tournaments formats. You can refer to this page to learn more about the various tournament formats.
"In a Constructed format, you build your deck in advance, using the cards in your collection. A Constructed deck must have a minimum of 60 cards, with no more than four copies of any card (excluding basic lands)."
In order of popularity, they are:
- Standard: Standard (formerly called Type 2, a long time ago) is by far the most popular constructed format. It's relatively easy to "buy into," since it only uses cards from the most recent sets, but players usually update their decks extensively to keep up with new set releases and rotation. The Standard metagame moves very fast because there's a major tournament every weekend, and numerous online events as well. Most Friday Night Magic events are Standard or Limited.
- Legacy: Legacy is the most popular "eternal" format, meaning that it incorporates cards from all of Magic's history. Legacy has a fairly extensive banned list to balance the format (which is what differentiates it from Vintage). New cards still make a big impact in Legacy from time to time, but generally the metagame is more stable. Much of the contemporary Legacy tournament scene exists thanks to third-party support from major Magic vendors like Star City Games.
- Modern: Modern is a fairly new tournament format. Like Legacy, cards never rotate out of Modern due to their age. Unlike Legacy, you can only play cards newer than Mirrodin block / Eight Edition (in most cases, an appropriate shorthand is "with the new card frame"). Modern is more accessible financially than Legacy. However, currently the tournaments tend to be a bit "seasonal" — many competitive players get excited about Modern when it's the format for PTQs or the Pro Tour itself, then ignore it for a few months afterward as they shift their attention to Standard or Legacy.
- Vintage: Vintage is the closest to an "anything goes" sanctioned format. Vintage players enjoy playing the most powerful, game-breaking cards in Magic's history. Costs of entry are prohibitive, however, and Legacy has a much greater following. (Many US tournaments allow some proxies, which greatly reduces the price barrier.) As far as I know, the best way to play Vintage consistently is to live in an area with a strong local community, but there are occasional large events like the Bazaar of Moxen as well.
- Block Constructed: Block Constructed has the smallest card pool of all Constructed formats, encompassing only the cards from the most recent block. Because cards are primarily designed for Limited and Standard, the Block Constructed metagame tends to be pretty stagnant and one-sided. With the exception of a small following on Magic Online, most people only really care about Block Constructed when it's the format for a Pro Tour.
- (Extended is dead as a doornail.)
Note that support for Legacy, Modern, and Vintage outside the US varies greatly by region.
"Unlike Constructed formats where you play with an already built deck, Limited formats require you to build a deck from a defined pool of cards. This pool might be defined by the contents of a few booster packs that you open at the event, or it may be from cards that you select in a Booster Draft."
The two Limited formats are (once more, in order of popularity):
- Draft: Booster draft using the most recent block or core set.
- Sealed: Sealed-deck Limited using the most recent block or core set.
Every Pro Tour includes a Limited component, and many Grand Prix tournaments are Limited. Limited is also very popular at the FNM level, at about the same level as Standard.
Rarely, a tournament will be a team event. These are essentially labels that modify one of the Constructed or Limited formats above.
- Team __________: Tournaments that just say "Team Standard" or "Team Sealed" involve playing individual duels, but players combine their scores to determine the team's overall standing.
- Two-Headed Giant: Two-Headed Giant is a specialized two-on-two multiplayer format.
(Most often, when players refer to a Magic "team," they just mean their practice buddies. Top players tend to prepare for tournaments with other top players, because it's the best way to hone your skills and try to predict the metagame.)
Non-sanctioned competitive formats
There are many Magic formats that are not officially approved for DCI-sanctioned tournaments. To my knowledge, the most popular non-sanctioned formats with serious competitive scenes are:
- Peasant/Pauper: A commons-only format. "Pauper" is the name on Magic Online, where it is most popular. (Peasant and Pauper are different because of how cards are "printed" on MTGO — for example, Hymn to Tourach is a common in paper Magic but only exists as an uncommon in MTGO.)
- Duel Commander: Also known as "French EDH," this is a modification of Commander for two-player duels. Most popular in Europe. It has its own official site.
It's also not uncommon to find multiplayer Commander tournaments at major gaming conventions, but these aren't really part of a large competitive circuit with a defined metagame.
Okay, but how can I find out what's popular in my area?
Wizards of the Coast has a Store & Event Locator feature on their website. Note that you can search by formats and event types; note also that it will mostly show game stores unless you filter out events like Friday Night Magic, Prerelease, and Game Day. When you click on a store, you'll see a schedule of their events, which will usually give you an idea of the formats they support. This list is only as good as the information that retailers feed back to Wizards' staff; the game stores do at least offer a reasonable starting point — you can visit a promising site in person and find out more information about what the local players are doing.
MTGMom is a calendar that lists major tournaments and qualifiers in the US and Canada in one handy page. Her list includes SGC and TCGPlayer events as well as WotC-sponsored ones.
For MTGO players, Magic Online has its own calendar page.
The popularity of the different formats is a tricky subject. First, it's very difficult to define. You could go by total number of players (is there a strong community?), number of tournaments (is it competitive?), available prize pools (is it serious?), size of tournaments (there's a difference between 10 tournaments of 8 people and 1 tournament of 80 people!), or any number of other metrics.
Secondly (and most importantly) this is not a constant thing across the world. Different places (even as fine-grained as different neighborhoods in the same city) have different popularity levels of different formats. For example, I live in Calgary, Alberta. Down in the south, there is little interest in Draft, but a lot in casual formats like EDH. There is very little for competitive magic, even for just winning packs at FNM. But in the north, there are a couple places that draft regularly with multiple pods with fewer casual events.
If you asked someone in the south what the most popular format was, they will probably say Commander or Standard. In the north, they're more likely to say Draft/Sealed and if you force them to say a constructed format, they're more likely to say Modern than Standard.
The bottom line is, I think you're asking the wrong question. What I would advise you to do is to ask at your local game store what formats are popular in your area. Consider looking for a secondary store to play at that might have formats you're looking for, too. Knowing what's popular nationwide or worldwide is going to be less useful to you.