SQL Compatiblity Mode

"Before you change the compatibility level of a database, you should understand the impact of the change on your applications. For more information, see ALTER DATABASE Compatibility Level (Transact-SQL)."
source: https://msdn.microsoft.com/en-us/library/bb933794.aspx

"Use compatibility level as an interim migration aid to work around version differences in the behaviors that are controlled by the relevant compatibility-level setting. "
https://msdn.microsoft.com/en-us/library/bb510680.aspx

Since, per Microsoft it is a temporary stop gap type measure until you can migrate to newer compatibility. I'd not do it while the db has anything going on, backup the db before starting, test it somewhere and be ready to modify any applications that may break. That's just from what I've read...not personal experience with it.

When database compatibility is set to previous version and they are attempted with procedure of newer version they will throw the Compatibility Level error. you need to change the Compatibility Level. Read this article to understand the affects of Compatibility Level on database

In addition to Andrew Jackson's good article that Jason Clark points you to above, please note that there were quite a few "breaking changes" when moving from SQL Server 2000 to 2005. There's a pretty comprehensive listing here. You should also see this page. There should also be an equivalent one for 2005 -> 2008
One example that comes to mind in particular is that it can be common to use a small query to provide a list for a combo-box: from 2005 it is not possible to use an order by clause within a sub-query, which broke a lot of code where I worked at the time.

Hope this helps

Mike