Q: Can I update my OS X10.4.11 to OS X 10.5.6 or new one  with this Macbook?

1) Make sure your computer meets Snow Leopard technical specifications - http://support.apple.com/kb/SP575  This mostly means making sure it is an Intel Mac and has at least 1GB RAM (at least 2 GB is recommended).  Go to the Apple in the upper left corner of any window to "About This Mac".  To check RAM continue to "More Info...".  It should say you have at least 1GB of "memory".

2) Buy a Snow Leopard disc from Apple online

http://store.apple.com/us/product/MC573/mac-os-x-106-snow-leopard - USA

http://store.apple.com/uk/product/MC573/mac-os-x-106-snow-leopard - U

or telephone Apple. In the USA: (800) 676-2775 (contact information for other countries: http://support.apple.com/kb/HE57)

Alternatively you can buy from online retailers but anticipate paying much more than the original price which is the price still charged by Apple

3) Backing up your computer before upgrading is always a good idea.

4) After installing from the DVD, update your Snow Leopard installation using Software Update or manually downloading and installing the Mac OS X 10.6.8 Update Combo v.1.1 - http://support.apple.com/kb/DL1399

If you are using old (PPC generation) software you may want to install Rosetta as well which I believe is an optional installation.

K

You can actually go up to Lion on that computer, as well.  Another option would be to purchase both Snow Leopard and Lion and place them in separate partitions, in order to be able to "dual-boot" and retain access to Rosetta for running PowerPC applications.

The final decision really depends upon your software and how you plan to use it; for example, do you link to an iPhone or other similar product that uses iCloud?

Can you tell us more?

Good suggestion about Lion. Requiring Lion requires Snow Leopard be installed first so there's no getting around that.  Also note that either OS may be happier with more RAM but the higher you go the more you need and with Lion you might seriously consider at least 4GB.

You could install Leopard but it is easier (and likely cheaper) to get hold of Snow Leopard. You will find a lot more software still available if you run Snow Leopard.

Some things to note, your Mac can upgrade to Leopard 10.5, Snow Leopard 10.6, and Lion 10.7.

However, there are advantages and disadvantages to each.

10.5's compatibility is closest to 10.4's, and includes Rosetta for compatibility of older PowerPC applications.

10.6 may have a few things not compatible, and Rosetta is an optional install to keep older PowerPC applications compatible.

10.6 offers compatibility with the latest Adobe Flash, and syncing iOS 6.  With 10.6, you can upgrade to 10.6.6 combo and then install 10.7, but not without the following issues.

10.7 No longer has Rosetta included, although someone can buy 10.6 server and Parallels to keep the compatibility of 10.6 and earlier. You can optionally also install 10.6 on its own partition for the occasional Rosetta compatibility you may desire.

10.7.3 offers the latest Java compatibility.

10.7.5 offers syncing with iOS 7, 8, and up to 9.1 with iTunes 12.2.25.  Unfortunately, the latest iOS 9.2 and 9.2.1 are not supported by your Mac's Mac OS X support, which means you have to be careful not to upgrade your iOS device through the nagging upgrade screen on your iOS device.

10.6 & 10.7 are far cheaper than 10.5, because of the availability of discs.

See my tips

http://discussions.apple.com/docs/DOC-2275 for 10.5 upgrade tips.

http://discussions.apple.com/docs/DOC-2455 for 10.6 upgrade tips.

http://discussions.apple.com/docs/DOC-6271 for 10.7 and above.