What is the distinction between chemokines, cytokines, interferons and interleukins?
Cytokines is the general class of molecules to which chemokines, interferons, interleukins and others belong. Biologists dispute whether something is a hormone or a cytokine, but generally the consensus goes with if it's to do with immunology it's a cytokine or if the resting concentration is in the picomolar range, but that's a very rough distinction. 

Chemokines are molecules that drive cellular chemotaxis. That means they make cells move towards a desired place. Generally chemokines refer to immune cells and there's loads and loads of them. 

Interleukins are anything which are messenger molecules between immune cells (inter- means between and -leukins means leukocytes/white blood cells). They're typically denoted by IL + number. However the interferon and tumour necrosis families come under interleukins too in most people's opinion. The interferons are a special group that typically inhibit viruses by making cells non-permissible to viral replication. They also do a few other things like activate macrophages or promote Th1 response, which both also interfere with viruses but there is bacterial overlap. The TNF family is a bit weirder as some are interleukin-like but others aren't as much. TNF-alpha (the classic one) is involved in macrophage maturation. However not all interleukins are strictly between white blood cells, as IL1 acts on the hypothalamus among other leukocytes. 

See the problem is everything is a bit grey because most cytokines (if not all) have more than one role and there's never consensus in the world of immunology when it comes to cytokines because more and more keeps getting discovered.