From my understanding, if I buy a piece of a real estate property in
the US and use it as my primary residence for over two years, then I
do not have to pay a capital gains tax when I sell this property. Now,
what would happen if this property was owned by my relative who lives
outside of the US? Would he have to pay a capital gains tax nomatter
when it was sold? What if my relative owned a company here in the US
that, in its turn, owned the property?

I am not an attorney. See the disclaimer below. A tax attorney is the
best source of info for these questions. The IRS is happy to answer
questions as well.

The property has to have been the sellers primary residence for two
out of the previous five years before the sale. Depending on how you
time things, you could sell a property every two years and defer
taxes. Other restrictions apply.

The relative outside the US is subject to more complex tax rules and
needs to consult a tax attorney.

If a company owns the property, then the company is taxed on the profits.