there's the common Internetional Phonetic Alphabet which is quite common on printed dictionaries.
Wordsmith uses a different kind of system(or at least that seems to me) when it presents the word of the day....e.g.(sord uhv DAM-uh-kleez) for SWORD OF DAMOCLES.
Has it got any specific name or it's only a clearer way to show the pronunciation of a word?

I don't know what it is called but I find it much easier than the IPA which I have never understood.

It's also less informative than the IPA and subject to misinterpretation.

I haven't met a printed dictionary that uses IPA yet, but I haven't met even half of them. ;-)

Here's a listing of Pronunciation respelling for English as used by different English language dictionaries. Hovering over the column headings will give the full name of the dictionary using that system, e.g. AHD = American Heritage Dictionary.