I replaced my internal drive in my macbook pro 2010 with a new SSD. I started the machine, then pressed alt/option to access Recovery Mode in order to use Disk Utility. So, I opened Disk Utility and tried to erase and partition the new SSD. This is the error I am getting: "Disk Erase failed with error: File system formatter failed". All the video that I have watched seemed to follow the same steps that I did. What am I doing wrong?
Your SSD as you said is empty, so this is why you do not have any recovery partition to reinstall the OS.
You need to boot to an external device (USB key or DVD depending on your Mac) in order to format and install Mac OS on your new SSD. You can use software like DiskMaker X to create a bootable OS X device.
You will need another Mac to create the bootable drive or you can reinstall your old disk, create the bootable drive and then reinstall your SSD.
Apple documents the steps for you and covers the options pretty well:
Basically, if your hardware support Internet recovery you'll be able to reinstall an OS and get a recovery HD on the SSD. If not, you'll have to bring a bootable drive to that Mac or create a recovery HD from another Mac.