I agree with Steven's definitions. I'll just add a few extra points:
For me, 'unit' has a more technical association. As with 'intensive care unit', it suggests that this is a specific treatment facility with appropriate equipment. 'Ward' is a more old-fashioned word, and it may simply refer to a room with beds in. For example, 'men's ward', 'women's ward', 'children's ward' - these are just sections of a hospital where particular in-patients are housed, regardless of the 'unit' which treats them.
'Clinic', as Steven says, is usually a small medical facility offering more limited treatment than a hospital. A facility which only treats skin conditions, for example, could be called a 'skin clinic'. Within a hospital or other facility, a 'clinic' doesn't refer so much to a specific location, but to a service which is offered there. For example, you could say 'There is an antenatal clinic held here every Wednesday morning and a heart care clinic every Thursday.'
These usages may be slightly different in different English-speaking countries.