Having a long position in a stock means that the investor is expecting to make money from a rise in the share price.
Investing is filled with esoteric terms that really mean simple things. This is one of them. When you are "long" a stock, that just means you own shares. That's it. Nothing bizarre. You expect to make money either from an increase in the share price or from receiving dividends or both. This is the "normal" position for most investors in the market.
In a long position, you have purchased the stock with the expectation that the price will go up in the future. The idea is that you will sell it in the future, hopefully for a capital gain. This is often a naked position that is not covered by some form of hedging. For most people, this is enough.
Just so you're aware, though, the potential loss for this position is 100% of the invested amount. That is, you could really mess up, buy the next Enron and watch the company go bankrupt, losing your entire investment. On the plus side, though, the potential gain is infinite, as you can end up buying the next Microsoft and continue to hold the position as the price rises and rises and rises... That's how people get those doubles, triples, and Lynchian 10-baggers -- by holding on as the stock price rises over time.