The recovery partition is what holds the windows recovery data from your system manufacturer. This enables their "restore to default factory settings", whatever they may call it. They do this so they don't have to ship a windows disc with the computer. Unfortunately, this will not restore any programs you've installed since you got the computer.
The system partition is what windows creates when you install it. This is where some system files reside. It's best to not mess with this partition.
The Acer partition is your C: drive. When they created this partition they gave it the name Acer instead of New Volume. Windows, Program Files and everything you have installed or copied or downloaded to the computer (before the new drive) resides on this partition.
Your new drive has been partitioned, it just has one partition on it. You partition a drive before you format it. You don't need more than one partition unless you want to break it up into smaller drives. Each partition will get it's own drive letter under My Computer. Some people like to name partitions for what they place on them (pictures, music, movies, games, recordings, etc) while others just like one large partition, the choice is yours.
Under Disk Management the diagonal lines you see on a partition just means that partition is selected. You can shrink or extend a partition size here, so if you decide to make more than one partition on your new drive, shrink the current partition to the size you want it to be and create new partition(s) in the unallocated space. Be careful though, if you shrink it smaller than your current data on it, you may lose some (never tried it myself).
A recovery partition does nothing for you here. There is no magic recovery by having a partition named recovery partition. If you're wanting some type of backup in place, I'd look into one of the RAID options that exist, or an external that backs up nightly.