They all seem to describe molecules of similar function and many people seem to use them interchangeably.
Also please include any other similar molecules if I've forgotten any in the list above.
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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.
In short, this is my understanding following my immunology course (which is mostly identical to AndroidPenguin's answer):
Chemokines: Produce cell movement, i.e. act as chemoattractants. Typically these are used to recruit more immune cells to a site of infection. (Name origin says it all: "chemotactic cytokine = chemokine...)
Cytokines: Produce an inflammatory response by altering transcription (via surface receptor signal cascades). These often act on non-immune cells.
Interferons: A type of cytokine. Secreted by virtually every cell in response to being infected by a virus. Produce an 'anti-viral' state in other cells, through MANY mechanisms. Causes the typical flu-like symptoms.
Tumor necrosis factors (specifically TNF alpha): A type of cytokine. Prepares endothelium (blood vessel walls) to support an inflammatory response by vasodilation, increased adhesion and increased permeability.
Interleukins: Some are chemokines (e.g. secreted by macrophages and dendritic cells upon activation to recruit further phagocytes and adaptive immune cells), others are cytokines (e.g. activating B cells to differentiate into plasma and memory cells after T cell contact).
Pharmacology cross reference: Recombinant type I interferons are injected as therapeutics. Viral infections would seem like logical indications, but interferons are both expensive and have considerable adverse effects, e. g., flu-like symptoms on injection, anemia and depression. Their application is therefore limited to life-threatening viral diseases, e. g. hepatitis C.