If you mean your iCloud Apple ID, you have to go to Settings>iCloud, tap Delete Account, then sign back in with your other ID. If you are signing back into the same account, your data will be synced back to your phone when you sign back in. If you are changing accounts (by setting up a new account with a new ID), when you delete the account be sure to choose Keep on My iPhone. Then create the new account, turn on your data syncing and choose Merge to upload your data to the new account.
If you change accounts you will be assigned a new iCloud email address. You will also lose your photo stream. If you have any photos in your my photo stream album that are not saved to your camera roll that you want to keep, before deleting your orignal account, open the My Photo Steam album, tap Edit, tap all the photos you want to keep, tap Share, tap Save to Camera Roll.