What are calories and how to burn them?
A calorie is a measure of energy. In nutrition, we usually mean the kilocalorie (kcal), which is the same as about 4.2 kJ (kilojoule). A average person needs about 2,000--2,500 kcal per day, but this of course varies quite a bit between people, depending on size (muscle mass, particularly) and level of physical activity (physical work, exercise).

Nutrition guidelines found on the labels of food products give a pretty good summary of the amount of energy they contain. I'm not sure what applications you have been using, but they most likely rely on the same basic data, so the values should be similar. You can use these values to figure out roughly what your calorie intake is, and to spot very calorie-rich foods. But counting calories exactly is actually quite difficult --- you need to carefully weigh everything you eat and keep track of lots of numbers, and it's easy to make mistakes.

Burning calories does not always mean burning fat. Fat is a long-term energy storage form in the body. When exercising, the body tends to burn stored carbohydrates (glycogen) first, and turns to burning fat when carbohydrates are exhausted. Therefore, prolonged, low-intensity exercise like long walks are usually better for burning fat than short "sprint" type of exercise. The body also burns fat during normal daily activity if your energy intake is lower than the daily need (when dieting). But it is also possible that some muscle mass is degraded (to amino acids) and used for energy as well.

See also Wikipedia articles on food energy and exercise. Please note that "how to burn calories" is a heavily debated issue, there is an entire industry of weight loss methods, and various claims about "easy" methods or products to lose weight quickly are generally not true. The only reliable method to lose weight is to reduce food intake and increase exercise, for a long period of time.