Applications are basically glorified ZIP archives. The java code is stored in a file called classes.dex and this file is parsed by the Dalvik JVM and a cache of the processed classes.dex file is stored in the phone's Dalvik cache.

An odex is basically a pre-processed version of an application's classes.dex that is execution-ready for Dalvik. When an application is odexed, the classes.dex is removed from the APK archive and it does not write anything to the Dalvik cache. An application that is not odexed ends up with 2 copies of the classes.dex file--the packaged one in the APK, and the processed one in the Dalvik cache. It also takes a little longer to launch the first time since Dalvik has to extract and process the classes.dex file.

Depending on the size of the odex/classes.dex file it takes longer to patch

Okay thanks. But I have applied patches even before I had busy box and it worked. 

Anyways thanks. 

How about. .odex: what does it mean or do? 

LP can't work without busybox because, busybox is "box" of tools, like tool for copy file (cp source destination), or create dirs (mkdir exampleDir) or compressing files and many mores.... It's easier for developers to use same tools from third part app instead of compiling everythink by themself (this tools needs tons of libraryies, and there are many problems with compiling them, its better use prepared tools, it saves time)... You can find Busybox for free on GooglePlay or just download some pirated version.