MacBook - Update memory = update software?

First we need to know which one of the 9 different models of MacBook you have. To see which model you have go to the Apple in the upper left corner and select About This Mac, then click on More Info. When System Profiler comes up check the Model Identifier and post it back here.

The Early 2006 model 1,1 Core Duo can only run a maximum of 10.6 Snow Leopard.
The models Late 2006 Core 2 Duos 2,1 through Early 2008 4,1 can only run a maximum of 10.7 Lion.
The Late 2008 model 5,1 Aluminum Unibody through the Mid 2010 White Unibody model 7,1 can run 10.8 Mountain Lion and 10.9 Mavericks.

You have:

OVERVIEW

Introduced May 2006

Discontinued November 2006

Model Identifier MacBook1,1

Model Number A1181

That model was introduced with Tiger (10.4). It can be upgraded to Snow Leopard (10.6). Check the Apple link below for requirements and ordering info.

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

Exactly as Stedman1 says..

The maximum Ram for Macbook1,1 is 2GB. If you have less I would increase that.

The model 2,1s, Late 2006 and Mid 2007 also shipped with 10.4. The Model 3,1 was the first to ship with 10.5.