Want to catch your own Latias? This is something that's easier said that done, so you'd better come prepared! This page will tell you everything you need to get a Latias of your own in each of the games.
Let's start with the easiest one: Sapphire. Beat the Elite Four and a level 40 Latias will begin roaming around Hoenn randomly. As soon as you encounter it... well, this is why you saved your Master Ball, right? I mean, you did learn something from Gen II, right?
...Go get a Pokémon with Mean Look and start watching your Pokédex. You can track Latias's progress across the map; just follow it and use Max Repels to find it. Latias has a catch rate of 3, meaning that if you throw a normal Pokéball when it has full health, you have a 0.4% chance of catching it. So bring someone who can put it to sleep and slowly knock down its HP, and eventually you'll have yourself a Latias.
(Unfortunately, due to a glitch, the Latias in Sapphire will have very low IVs. Sorry.)
You can get Latias in Ruby if you're a timetraveller or if you have a Gameshark. You need an Eon Ticket to catch Latias in Ruby; the Eon Ticket allows you to travel to the Southern Island, Latias and Latios's home. There, you can fight a level 50 Latias one-on-one (allowing you to get a Latias with the nature you want) and avoiding the low IV glitch. The problem, of course, is getting an Eon Ticket. If you didn't get it back in 2003, you're not getting it now.In Emerald, after you beat the Elite Four, your mother will ask you what color the Pokémon on the TV was. If you answer red, Latias will roam around the world, while Latios will remain on Southern Island. Pick blue, and vice versa. The other Pokémon is then only accessible if you have an Eon Ticket.
While not actually native to the games in this generation, Latias and Latios were made available in them. Of course, once again it mirrors their original release.
Latias is native to HeartGold, and starts roaming after you find and speak to Steven Stone, who's hanging out in Silph Company in Saffron City. It'll be at level 35. To get Latias in SoulSilver, you need the Enigma Stone, which was distributed via Wifi in Japan and North America in 2009 and 2010. You'd have to take the Enigma Stone to Pewter City, where it would be revealed that it was really the Soul Dew, and a level 40 Latias would attack the player. Tada, Eon Pokémon!
This is the generation I never actually played, so apologies if there's anything off here.
When you enter the Dreamyard in White 2, you'll encounter a level 68 Latias. Latias will lure you around the area, and eventually it'll let you attack it so you can capture it.
Omega Ruby and Alpha Sapphire actually change up the way to get Latias and Latios in that you don't have to go through the whole roaming nonsense with them anymore. In fact, you don't even have to catch them — as soon as you surf to the other side of Route 118, your version's Eon Pokémon will appear before you and whisk you off to Southern Island. A level 30 Latias will promptly join your party in Alpha Sapphire; Latios in Omega Ruby.
Getting the other one, then, is just as much of a hassle as it always has been. You still need the Eon Ticket, but instead of just downloading it from the internet, Nintendo decided to give it out over freaking Street Pass. That's right, you have to randomly come across someone who has the Eon Ticket in real life and hope it gets passed to your game. This is not helpful, Nintendo. This is just as frustrating as it has always been.