Sometime after you beat Story Mode this dungeon will be available for exploration. It is reasonably important if you want to catch all the Legendary Pokémon, as it is home to Deoxys. It is a very long and arduous dungeon, and there is one major catch; you can only bring 3 items into it! Actually, you can technically bring 4, as a hold item does not count. That said, a Stamina Band or X-Ray Specs along as a hold item. The Stamina Band will help slow your Belly depletion, which is going to help a lot as this dungeon has 20 very large floors. Details on the dungeon later.
The X-Ray Specs will help with finding Deoxys on the floor. Or, rather, a mirage of it. To make the stairs on each floor appear, you will have to find (sometimes a challenge) and battle Deoxys' mirage. The X-Ray Specs will help you find him so you can battle him. However, you may want to switch between Stamina Band and X-Ray Specs if you can, so your Belly will not deplete.
This dungeon will be very easy, however, if you managed to find an ultra-rare Tight Belt. This hold item completely stops your Belly from depleting, which means you can focus on finding and beating the Deoxys mirage on each floor. But as mentioned, it is very rare, and it is almost always only found in walls.
Your other item(s) should include a Reviver Seeds if your lead Pokémon has a lot of HP. If it does, go through the floors until your Belly runs out, then continue to let it run out as you walk which, if you have a lot of HP, will be a little while. Then faint, and you will be revived and ready to start again. If you do not have a lot of HP, then simply bring a Huge Apple or two along, and hope they last you through the 20 floors.
Battling Deoxys
If and when you exit the 19th floor, you will see Deoxys in the middle of a room. He will say how he woke up one day and found himself inside the cave. He will demand that you tell him how he got there, and then battle ensues.
Deoxys is a very tricky Legendary, for a few reasons. The first is the fact that he is incredibly hard to defeat in the first place. He has around a whopping 1,000 HP, but some relief is the fact that his only real attack is Night Shade, which will do 45 damage. Another reason he is tricky is the fact that his recruit rate is approximately -34%, so the only way you will have even a reasonable chance of catching him would be to have a Lucario Rank, and a Friend Bow equipped when you defeat him. All in all, a very tough fight.
Also, Deoxys gives you Enclosed Island as a Friend Area when you recruit him; it cannot be purchased. Deoxys will be at level 45 when you recruit him.