How do I completely remove Ubuntu 12.04 on my Mac?

Since you're switching back to Ubuntu 10.04 LTS, you won't want to expand your other OS (I am guessing it's Mac OS X) to fill up the space that had been used for Ubuntu's partitions, because you'll just be installing 10.04 after removing 12.04. This simplifies things.

It is a good idea to actually remove the partitions though, to ensure that Ubuntu 10.04 LTS's installer sees unpartitioned space that it can use.

Boot from the Ubuntu 10.04.4 LTS installation CD/DVD or USB flash drive. (It's worth getting that point release's installation media even if you have the original 10.04 LTS installation media or media for an earlier point release.) Select Try Ubuntu without installing (not Install Ubuntu).

Make sure to back up any data in your Ubuntu partitions, as you'll be deleting them and it's a hassle (and sometimes impossible) to recover data from a deleted partition. It's also a good idea to ensure your backups of documents and other important files in your other OS's partition are current. This is less necessary since you won't be resizing that partition, but you could still make a mistake.

Open GParted (System > Administration > GParted Partition Editor). The Ubuntu partitions are of type ext4 and linux-swap (or, if you started with a very old Ubuntu system, ext3 and linux-swap). Remove them, but make sure to keep other partitions. (These other partitions are probably, but not necessarily, of type hfs+, since that's what Mac OS X uses. If your other OS is Windows, then its partitions would probably be ntfs.)

After removing those partitions, click the green check mark to apply your changes. Wait for it to finish. Then quit GParted. Double-click on the Install Ubuntu icon on the desktop, and this will start the Ubuntu 10.04.4 LTS installer. When you install, make sure not to tell it to use the whole disk, as this would wipe out your existing (Mac OS X?) operating system.