What is the name of the part by the right-hand pedal on to which the chain grips?
The entire assembly is called the crankset.

The ring with the teeth is called the chainring.

The crankset consists of the crank arm, the spider and the chainrings.
The chainrings bolt onto the spider which is the multi-armed piece connected to the crank arm. 

The right crank arm and spider are often one piece. Lower-end cranksets are forged or riveted together, while higher-end (like the one shown in the question) are bolted. With bolted cranksets one can exchange only the chainrings; with monolitic ones the whole assembly has to be replaced once the chainrings are worn.