This: цей (masculine) / ця (feminine) / це (neuter)
That: той (masculine) / та (feminine) / те (neuter)
In the example you gave, "this cat" is accusative, as in the question "whom?". So here it would be "Я можу погладити кота?" — "Якого кота? Цього кота?" This is a bit too early for you in this lesson, but I felt like I should bring this up for more advanced, but not yet fluent learners who might look here.