What are the tests that police perform to determine that someone is driving under the influence?

SFST. Standardized field sobriety test. It is a series of test that include the walk and turn, nystagmus, and the one legged stand. All the other tests like reciting the alphabet or counting are just extras.

Additional: They also use screening instruments known as Alka-Sensers (known in police slang as 'blow sticks') to determine if the exhaled breath contains enough alcohol to create reasonable suspicion to warrant the use of the SBST and/or a formal Breath-Alyzer test.