A ward is just a name for a section of a hospital , for example, the psychiatric ward. A unit basically just another name for a ward, although it's generally only used with certain areas, such the intensive care unit. A clinic is the name usually given to a medical facility that is smaller that a hospital, but it is not part of a hospital itself.

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.