Programming is a very demanding task, and it requires a special kind of person. Most people who try their hand at programming decide that it is either too much work, too boring, or a combination thereof. Fortunately, these people are the ones you don't want to be programming, so it all works out.
Assuming you are not one of those people, there are a few key indicators that tell you whether you are cut out to be a programmer. The first indicator is your mental aptitude. Do you enjoy solving problems and puzzles — for example, a complex math problem — and do you like the more technical things? Programming at its core is about solving problems and figuring things out, so if you don't have the right mental aptitude, then programming probably isn't for you.
Related to the first indicator, the second indicator is your willingness to work through difficult problems and figure things out. This skill requires a very dedicated, almost obsessive person who is transfixed on the problem. Even if you don't know something, you need to be comfortable with that feeling, and still be able to rationally research a solution to the problem.
The third indicator is your passion for learning and growing. Do you spend hours at a time reading and researching in books and documentation? When you see high-quality programs and games, does that make you want to dissect the program to find out how it works? Passion is often what drives you to keep pushing to the next level, and without it, you will almost be guaranteed to just get tired and give up.
All three of these indicators — aptitude, persistence, and passion — are vitally important to any successful programmer. We all face challenges and difficulties, but we overcome them using those different things. If you only have two of the three indicators, you can still be a decent programmer, but you won't be as complete a programmer as you could be.