What is the American version of Australian Tafes?

I think it's a community college but I get the impression that they're more like B grade universities.

Until relatively recently vocational-technical schools were fairly common in the USA and they were separate from the university system. They were state-government funded institutions that taught subjects like welding, plumbing, and other trades.  Students taking a course of study at a voc-tech school did not have classes outside the subject of the course. Someone learning how to be an electrician would not take classes in subjects like biology, English, or history.  The school also did not offer those classes.

At the same time there were "junior" colleges which were part of the university system and which offered Associate's degrees (two-year programs) in the regular subjects offered at a full university.   In some states, such as Florida, students were strongly encouraged to take their first two years of university at a junior college in their hometown and then be a boarding student at a full university for the second two years. It was done partly to reduce complications since if someone is going to drop out of university they are most likely to do it before the end of their second year. Having them attend non-boarding junior colleges weeded out the ones who were going to drop out, thus reducing the student load and increasing the success rate for the full universities.

The trend has been for state governments to combine the voc-tech schools with the junior colleges.  The combination re-focuses the requirements so that welding students must also take classes in academic subjects to graduate, though many of them still offer "certificate" programs (no academics) and not just "diploma" (academics required) programs. 

These are examples from where I live:

http://www.dcc.edu/

https://www.mybrcc.edu/

https://www.rpcc.edu/

At the same time, private training academies that teach mainly vocational and technical courses have come into existence which cater to people who don't want to spend the extra time taking biology, etc. when they want to learn to repair electronics/whatever.

These are examples:

https://paulmitchell.edu/

https://www.remingtoncollege.edu/

https://www.brsc.edu/programs

Note the private schools can offer Associate's degrees, but one must be careful as their degree programs tend to not be nationally accredited.  

Generally in the US that would fall under the community college category.  Some states have something like education cooperatives that teach a variety of skills including trades.  NYand Colorado have B.O.C.E.S.  I'm sure there are similar programs in most states

A TAFE is a school that teaches  Technical and Further Education. It fits in the gap between  high  schools  and  Universities. The qualifications that  you get at TAFE  will get  you a job  but can  also be  rolled over into  Credits  at  University  if  you decide  to keep  going  with  your  education.

I have no idea  what the American  equivalent is

If you explain what a Tafe is, we might be able to answer. 

Community college I'm guessing