Don't you just hate that? I just keep mashing buttons until something happens on mine!
Hibernation is where the system saves the contents of the memory to your hard drive before shutting down (to conserve power). You also might be experiencing "Standby" - which is where the system leaves a trickle of power in the memory to save it's contents. The noticeable difference is when you turn it back on - Standby comes on almost immediately - whereas Hibernation requires longer to boot back up (and it usually says "resuming from hibernation" when it does.
To further confuse the issue, your BIOS has different power states that resemble these functions - you'll see them in the BIOS most commonly as S1 and S3 states.
To stop it from doing this, it depends on how the laptop is setup by the manufacturer. Here's the Windows way to do it - if it doesn't let you do it, then it's got some manufacturer stuff on it that you'll need to adjust instead.
Go to a blank area of your Desktop and right click on it.
Select the ScreenSaver tab
Select the Power button
In the next screen should be (on one of the tabs) options for Standby and Hibernation - they also may have options for AC power and for battery. I'd suggest leaving the battery settings as they are - and then turning off the options for when you're on AC power.
If your laptop is taking a long time to load, I'd suggest looking at this thread for some ideas on how to speed up the loading: http://www.bleepingcomputer.com/forums/t/44690/slow-computer/
While I don't have problems with leaving a desktop always on - I shy away from doing this with a laptop due to the heat issues. Maybe you'll want to use "Standby" with the laptop to help speed up the loading.
While there's several ways to turn off indexing, this'll probably work best:
1) Go to My Computer and right click on each hard drive and select Properties.
2) At the bottom of the General tab will be a check box for the indexing service. Uncheck it, and if you get a pop up, tell it to apply it to all sub-directories, etc. Do this for all hard drives.
3) Go to Start...Run...and type in "services.msc" (without the quotes) and press Enter.
4) Scroll down until you locate the Indexing Service, right click on it and select Properties.
5) Use the dropdown box that says Statup type and select "Disabled". Then click on the Stop button below it, then click on OK to accept the changes and exit the dialog.