Indeed they are bacteria! The Gram stain is a common, inexpensive, and very useful test that Medical Technologists (laboratory workers in hospitals) do to help doctors diagnose and treat bacterial infections.
The bacilli part simply means the shape of the bacteria. A bacillus is a rod-shaped bacteria - think of a tiny sausage! The other common shape is a "coccus" or "cocci" (cocci and bacilli mean there is more than one) which is a sphere - like a tiny grape.
I won't go into the details of the test, but basically the specimen (whatever body fluid was submitted to the lab) is stained in a special way that typically shows one of two different results: Gram negative or Gram positive.
Why does Gram positive or negative matter? First, it helps with deciding WHAT the bacteria is and second, it helps determine WHAT ANTIBIOTIC can be used to treat an infection with it.
Gram positive bacteria have a very thick outer cell layer made of peptidoglycan which stains dark purple. Gram negative bacteria have different layers on the outside of their cells and the peptidoglycan layers are much thinner - so they stain pink.
So a Gram positive bacillus is sausage/rod shaped that stains dark purple with a Gram stain test.
Examples are: Clostridium, Bacillus, and Listeria, but there are MANY more!