This is a question I have been fielding a lot lately. The most common scenario was that you were sitting at home, visited your trusty website that has been acting fine for the last two years, and all of a sudden it's broken! It may be that the reason you are seeing this error message is because your host has upgraded your account to include the latest version of PHP. In the release of PHP v5.3 many of the functions that have been slowly dying out have finally been deprecated and replaced with others and are not backwards compatible. Most certainly, if you are running Joomla 1.0 you are likely to get quite a lot of this. There are also quite a few Joomla extensions (plugin, components, and modules) that will be affected as well.
The one I seem to see most often as of late is the split() function. If you see the following message:
Deprecated: Function split() is deprecated in /home/xx/xx/xxxxxx.php on line 60
This means that split is no longer being used and will need to be replaced with a function that is.
If you know a little about PHP you can do one of two things:
1) Replace the split() function with the new function preg_split. Be careful though, the expression following will need to be correctly formatted for this to work.
2) Replace the split() function with explode(). This is probably the easiest of the two methods and tends to work just fine as a replacement.