You must upgrade to Snow Leopard 10.6.8 to access the App Store. You also must have an Intel Mac.
Upgrading to Snow Leopard, Lion, or Mountain Lion
Upgrading to Snow Leopard
You can purchase Snow Leopard by contacting Customer Service: Contacting Apple for support and service - this includes international calling numbers. The price is $19.99 plus tax. You will receive physical media - DVD - by mail.
Third-party sources for Snow Leopard are:
Snow Leopard from Amazon.com
Snow Leopard from eBay
After you install Snow Leopard you will have to download and install the Mac OS X 10.6.8 Update Combo v1.1 to update Snow Leopard to 10.6.8 and give you access to the App Store.
Before upgrading check that you computer meets the minimum requirements:
Snow Leopard General requirements
1. Mac computer with an Intel processor
2. 1GB of memory
3. 5GB of available disk space
4. DVD drive for installation
5. Some features require a compatible Internet service provider; fees may
apply.
6. Some features require Apple’s MobileMe service; fees and terms apply.
Upgrading to Lion
First, you need to upgrade to Snow Leopard 10.6.8 as stated above.
You can purchase Lion by contacting Customer Service: Contacting Apple for support and service - this includes international calling numbers. The cost is $19.99 (as it was before) plus tax. It's a download. You will get an email containing a redemption code that you then use at the Mac App Store to download Lion. Save a copy of that installer to your Downloads folder because the installer deletes itself at the end of the installation.
Before upgrading check that you computer meets the minimum requirements:
Lion System Requirements
1. Mac computer with an Intel Core 2 Duo, Core i3, Core i5, Core i7, or
Xeon processor
2. 2GB of memory
3. OS X v10.6.6 or later (v10.6.8 recommended)
4. 7GB of available space
5. Some features require an Apple ID; terms apply.
Upgrading to Mountain Lion
Be sure your computer meets the minimum requirements:
Apple - OS X Mountain Lion - Read the technical specifications.
Macs that can be upgraded to OS X Mountain Lion
1. iMac (Mid 2007 or newer)
2. MacBook (Late 2008 Aluminum, or Early 2009 or newer)
3. MacBook Pro (Mid/Late 2007 or newer)
4. MacBook Air (Late 2008 or newer)
5. Mac mini (Early 2009 or newer)
6. Mac Pro (Early 2008 or newer)
7. Xserve (Early 2009)
Are my applications compatible?
See App Compatibility Table - RoaringApps - App compatibility and feature support for OS X & iOS.
Am I eligible for the free upgrade?
See Apple - Free OS X Mountain Lion upgrade Program.
For a complete How-To introduction from Apple see Apple - Upgrade your Mac to OS X Mountain Lion.