The Tamron AF 70-300mm f/4.0-5.6 Di LD Macro lens is compatible with the Canon 600D or T3i. The Di designation actually means that it is compatible with both full frame and crop sensor cameras such as the 600D.
It is termed "macro lens", so yes it will do macro photography, but it is not a true 1:1 macro lens. It has a maximum magnification in the macro mode of 1:2, and only 1:4 without the mode. What this means is that the classical term "macro lens" does not apply to this lens. Instead, it has the capability to achieve a large subject size, but not quite what a dedicated macro lens can achieve. I would consider this lens to be very useful in macro photography though.
If you are looking for a lens under $200 in this range, that also does macro to a degree, this is a viable option and will get it done for you.