If you got actual results from the first allergist, do bring it up. Otherwise just explain your symptoms and what you think you MAY be allergic to so they can test you for that.

As for what happens at an allergist, it really depends on the tests done.
I've had scratch tests (they put a drop on your skin of allergen, then use a needle to agitate the skin so the allergen gets through the skin barrier) and I had this other test where they used something that looked like a HUGE Lego block that had different allergens in the different needles and they flip over this thing onto your arm and all of the needles enter your skin basically at the same time. Surprisingly not as painful as you might imagine.