I have heard alot about the new "cheaper" laptops that are going to be out on the market soon. How much RAM does one have?

If you are referring to netbooks then the short answer is "not much." Netbooks generally have between 512MB to 2GB. Most new laptops on the market now start at around 2GB of RAM and you can commonly buy one with 3Gb or 4GB for less than $500. Netbooks are popular more for their reduced size and weight. The average netbook will have a 10.1 inch screen, the average laptop will be around 15". Netbooks also generally have slower processors which use less power, but also, obviously, reduce performance. Most netbooks are still sold with Windows XP because their systems specs fall below Vista's requirements. The new Windows 7 operating system is supposed to correct this problem and you should start seeing more Windows 7 equipped netbooks this holiday season.

The ASUS Eee PC 1005HA-PU1X-BK at Amazon has the following:

    * Up to 10.5 Hours of Battery Life and Free Sleeve Case Included
    * Intel Atom N280 Processor 1.66GHz
    * 1GB DDR2 RAM, 1 x SODIMM Slot, 2GB Max; Wi-Fi 802.11 b/g/n
    * 160GB SATA Hard Drive (5400 RPM) and 10GB Free Eee Storage
    * Windows XP Home Operating System, 10.1-inch WSVGA 1024x600 LCD Display

It sells for about $350.

For the same price you could get an Acer from Best Buy with:

# AMD Athlon™ single-core processor
Features 667MHz system, 512KB L2 cache and 1.6GHz processor speed.
# 3GB DDR2 RAM
For multitasking power.
# Multiformat DVD±RW/CD-RW drive with double-layer support
Records up to 8.5GB of data or 4 hours of video using compatible DVD+R DL media; also supports DVD-RAM.
# 15.6" WXGA high-definition widescreen display
# 160GB SATA hard drive (5400 rpm)
# Free upgrade to Windows 7

I am not sure exactly what you have heard but there is some cheaper laptops out there now and depending on how cheap you want to go how much RAM you are going to get.  Also what service you want.   I have a  dell and they start at 399.99.  If you don't add any extras to it that would be your price and they include a year of service so that if you have any problems your computer is fixed.  I paid a little extra (but not much) for inhome service so if my laptop would happen to break they come directly to my house to fix it.  I work from home so being able to get it fixed very quickly is important to me and sending it in would take way too long.  Also not sure what you are needing the laptop for but another option is a mini.  They are great for internet surfing, watching movies, checking emails.  They also have a longer battery life but I would not recommend one if you have a lot of work to do as the keyboards are much smaller.  They have a 1gb RAM and honestly that is all my laptop has and I have not had any problems with it but would like to have 2gb for  my new one. 
One Mini that you might want to take a look at is the Dell Mini 10v.  It has 1gb memory and comes with Windows XP and wireless connection.  For a cheap Dell Laptop I would look at the line of Insprion 11.  It has Vista Home edition and 2 gb and also wireless. 
Another deal you may want to look at is the Toshiba I have attached.  This is a very good deal on a laptop with 3gb and it also has a webcam.  Good luck with your new machine and thanks for using IMShopping!