What is static charge?

Static charge is nothing but accumulated charge on surface of the body. For instance in all the example given in your question - be it rubbing the ballon, or a door handle - the charge is accumulated on the surface that results in such interesting experiences.

Lets dig into the reason behind this charge. You would have studied in your class that all bodies are composed of atoms. And Atoms have three parts - neutrons, protons and electrons. Neutrons have neural charge, protons are positively charged and electrons are negatively charged.

Normally, a body (sometimes called matter) is charge - neutral. That is, it has equal number of electrons and protons. So if p is the number of protons and e is the number of electrons, p-e =0. If p-e < 0 that means it is negatively charged because it has more electrons than protons and vice versa.

Now what causes a neutral matter to get negatively charged or positively charged - some kind of activity like rubbing a balloon. electrons can move from one matter to other and thus creating the imbalance and hence the charge. This imbalance result in the static electricity. You would know that opposite charges attract while same charges repel.

Try this experiment - take a pencil and rub it with your hairs and then try to bring it close to small bits of paper. You can see the live demonstration of the static charge.