The simple answer to this question is no; eating fatty foods will not make you fat. The more complicated answer is maybe.
If you eat more than you burn, your body will store those extra calories as fat and you'll gain weight. If you eat less than you burn, your body will draw on previously stored fat for energy and you'll lose weight. If you eat the same amount of calories that you burn, your weight will remain the same.
Fat is a macronutrient just like carbohydrates and protein. It is more energy dense meaning it contains more calories per gram than carbs or protein but other than that, there's nothing special about fat that will cause you to get fat if you eat it.
Because it is more energy dense, cutting fatty foods from your diet will reduce your calorie intake a lot more than cutting carbs or protein from your intake. Fat however, does has some essential functions in the body and completely eliminating it from your diet is a bad idea.
The Bottom Line
Remember, at the end of the day, weight balance is about the total amount of calories you eat, not just the ones you get from fat, protein or even carbs.