The MyBB software has a converter designed to migrate the content from other supported forum software to MyBB. Currently the scripts that can be migrated to MyBB are:
Invision Power Board 2.1, 2.2, or 2.3
MyBB 1.4 (Merge multiple copies of MyBB in to one)
vBulletin 3.6 or 3.7
The migration procedure is not hard to accomplish still there are several requirements:
You need to have MyBB already installed under your account.
The forum that you are migrating should be under the same account as your MyBB installation.
You need specific details about the forum you are migrating from (in all cases you need access to the old forum database, thus you will need the database username and password and the database name itself).
You need to download the convert module for MyBB and upload it under the MyBB installation
It is highly recommended to always back up both forums (the MyBB and the one you are migrating) prior to initiating the migration. This way you will ensure you can revert your forums back to a working state in case anything goes wrong.