The basic position is that copyright exists in virtually every work of art, for a very wide definition of 'art', and belongs to someone or some organisation and to use a work of art produced by someone else you need at least their permission. If you intend using it commercially the copyright owner may be able to demand a fee.
If you are going to use an image like this on a web-site it would be at least courteous to ask the creator's permission before using it. For non-commercial purposes this may well be freely given. As an example, there are quite a lot of my photographs scattered around the web but in each case that I know about my permission was given and I get a credit for them.
In practice, given digital technology, it is virtually impossible to stop someone using your work if they want to, but asking permission and giving a credit will save you from the possible embarassment of getting a DCMA notice.
This is a topic which has been discussed at some length in two topics that I know of in the SpeakEasy section, see
There may be more.