Pokémon Emerald, like all other Pokémon Games has it's fair share of Legendary Pokémon and One Off Pokémon and here is a list of the ones we know of

The First One Off You are likely to see is Rayquaza, Apart from heading to the Sky Pillar to awaken it before Sootopolis, you have to catch it after that. It is Level 70 still so be prepared
The next you are likely to see is in the Battle Frontier. In the furthermost South East section, you will see a strange looking tree. If you use your Wailmer Pail on it, it will awaken as a Level 40 Sudowoodo
Next would be Groudon. Found on Routes 114, 115, 116 or 118 After beating the Elite 4 and talking to the Weather Institute Owner. Have to get to it in time though. It is in the Terra Cave and is at Level 70
Next would be Kyogre. Found on Routes 105, 125, 127 or 129 After beating the Elite 4 and talking to the Weather Institute Owner. Have to get to it in time though. It is in the Marine Cave and is at Level 70
Next would be Mew. Found on Faraway Island, it spots you and plays a game of Hide & Seek with you, once you find it, it will Jump out and attack you. It is Level 30 So Be Prepared
Next would be Lugia. Found on Navel Rock with the Mystic Ticket. Once you get there take the right route and go downwards 12 floors. It is there at Level 70
Next would be Ho-oh. Found on Navel Rock with the Mystic Ticket. Once you get there take the left route and go upwards 6 floors. It is there at Level 70
Next would be Deoxys. Found on Birth Island with the Aurora Ticket.. It is Level 30 So Be Prepared
Next would have to be Latios & Latias. You can only get one of these in this game. After you beat the Elite 4, you will see a report on the news about the Mysterious Pokémon. After the Show ends, your mother will ask you about it asking what colour it was. Blue or Red. Pick the Colour you want and that'll be the Lati@s
After That is Regirock. Open the Caves this Way but the way to access the part of the cave is different. For Regirock in Route 111 you have to read the Braille and go Left, Left, Down, Down and then use Rock Smash
After That is Regice. Open the Caves this Way but the way to access the part of the cave is different. For Regice in Route 105 you have to read the Braille and and walk around the cave once
After That is Registeel. Open the Caves this Way but the way to access the part of the cave is different. For Registeel in Route 120 you have to read the Braille and go to the center of the Room and use Flash