System (Required if you maintain Win7)
This is just a 100mb partition Windows puts at the front of their installs (in 7 at least), which contains stuff like bitlocker (if in use) and some boot pieces and recovery piece (enough to get system restore to function, anyway).
C (Required if you maintain Win7)
Obviously this is your Install, meaning it is the operating system of Windows 7 (currently) and then your user files (C:\Users[username])
HP_RECOVERY (optional, could be backed up to DVDs)
This would be the equivalent of recovery CDs that use to ship with computers, so if W7 dies (or you dislike W8), you could theoretically restore from this.
HP_TOOLS (Optional, probably backup-able)
Generally these are their diagnostic tools, so if you call the service center to say my computer is acting funny, they will encourage you to boot into that and run diagnostics to see if any parts are deemed "bad" before they have you ship it in.
In short, which are safe to delete: That is really up to you. How badly do you want the ability to restore and run Diagnostics? Did you back up your User data? If it is no, yes (respectively) and you no longer want W7 installed - you could delete them all. However, if you want to keep 7 installed, do not delete the System Reserved partition.
Unfortunately, Windows (maybe 8 has changed this..?) can only function on "Primary Partitions", so you will have to remove something if your intention is to dual-boot.
One option for you (if you have the space on an external hard drive) is to backup the disc as an image (with Norton Ghost, Acronis True Image, or DD on a linux live disc -- I am sure there are free options in Windows, but I am unaware of them right now).
Or, just back up the Recovery stuff to DVD's per HP's guide.
If you are going 100% Windows 8
I would do the following:
- Backup the Recovery partition to DVD's per the HP's Guide (link above).
- Boot a Linux Live Disc and make a bit-for-bit copy of the tools
dd if=/mount/externalHD/HP_Tools.iso of=/dev/sda4 (Please research DD before performing this)
- Backup your user data (however you prefer)
- Trash all partitions and start fresh.