If you want something cheap/used and works good, go for the oldschool Thinkpads on ebay, I got a model T400 which included 2GB or RAM that you can upgrade to 8GB. The CPU is 2.26GHz which is decent for doing homework and maybe playing some flash games.
If you're planning on getting a newer model or something completely different, look at how many cores the CPU has, it's GHz power (make sure it's above 1.9GHz or 2.0GHz). If the CPU is intel, make sure it's post Sandy Bridge design (The 2005 design, you need to check this if you're getting a used one). Also don't forget the clock multiplier. Newer laptops will have better clock multipliers so no worries over there.
My main laptop is a Lenovo G580 which I love, it comes with 4GB of RAM which you can upgrade to 8GB by adding another one. Make sure that you check its Hz to match it.
MAKE SURE IT MATCHES OR IS BELOW YOUR CPU's Hz!!!
My tip is: Don't buy consumer crap like touch screen nonsense if you're planning on doing more than one thing at a time. These usually cost a lot and not only that, they have terrible hardware:
1. Small screen resolution
2. Small disk space (probably running off an SSD if it's a small laptop/tablet hybrid)
3. Low CPU power (under 2.0GHz)
4. Low RAM (2GB max. Laptop/tablet hybrids cannot be upgraded as they are soldered with the motherboard)
But I'm not saying these are 100% bad, if you delete the bloatware programs and use a browser that runs smooth like Opera it will work pretty good on a 1.3GHz CPU, 2GB RAM tablet/hybrid. You just gotta get the right things for what you're planning on doing on the computer. These might work good for doing homework and such, but nothing extreme like playing a Counter Strike game.
Make sure: If you're buying something recycled off ebay that most of the parts will not come with it, which you can order off amazon (to make sure you get new parts that is) The thinkpad I got did not come with a hard drive, battery and the adapter (which I had one). So the total was around $120 maximum. I did not bid on the laptop but straight just bought it for $42, and with the shipping it was $55 alone. The battery and hard drive cost $13 and $46 respectively. The laptop works great after installing Debian Wheezy, I'm mostly doing homework and programming on it.
Note: You can message the seller on ebay and ask if they clean their laptops they are selling. I asked if I should clean it with alcohol and they responded saying that their products are not filthy and that they already have cleaned it with alcohol. I only had to buy a new trackpoint (the little red thing in the middle of the keyboard) because it had some black marks on it and I wasn't comfortable around using that.
If you are planning on cleaning your keyboard with alcohol (if it's used) make sure the alcohol is a 99% isopropyl alcohol because it's better than having lower %s, but usually you just gotta wait longer for it to dry if it's 91% if I recall correctly. Here is a video on how to do it correctly:
https://youtu.be/T7za8WpqVbQ?t=2m10s
I'm sure there is more to this, for example, depending on your budget, but I'm guessing buying a used one off ebay is the lowest-budget way to go, but if you want something new and the budget isn't your thing, then of course getting a new model is nice. Just make sure it's not a consumer crap marketing design.
Ask if you have any questions. I know there is more to this.
Edit: oops, this was a dead thread? ;_;