The London Stock Exchange is a system which allows people who want to sell shares and people who want to buy them to connect. In fact, you could compare it to Ebay as Ebay doesn't sell anything it just connects sellers and buyers.
To buy or sell on The Stock Exchange you need a Stock Broker, this could be a website or a real person who facilitates the sale or purchase of shares. In the early days, UK stock trading took place on the Stock Exchange Floor but these days it's done electronically.
Stock exchanges are global, there are many markets around the world and within those markets there are sub-divisions. It's not that complicated though, in the London Stock Exchange there is the main market and the alternative investment market (AIM), then there are specialist and professional securities markets. They all work the same way though and the only difference in how you buy or sell on them is whether or not your broker (or share dealing website) will let you access them.
The FTSE 100 Index is the Financial Times Stock Exchange Index and it shows the 100 most highly capitalised companies on the London Stock Exchange. These are the companies which people most often buy and sell shares in. The global and national markets have different names, which can be some of the acronyms you'll see bandied about and often businesses within those markets are also referred to by acronyms for short. If you don't know what something stands for a quick online search should reveal which group of companies is being referenced.
You might say that Natwest Group, or Barclays, or Sainsbury's was down, meaning that their shares were being sold for less (worth less) than they had been previously, equally you might say that any of these companies was 'up' and you'd say by what percentage their share had increased or decreased in value.
To make money on the stock market people need to buy shares when they're at their cheapest (down) and then sell them when they're worth more (up) to make a profit. The middle men in the UK market who are in between stockbrokers of those selling and buying are called 'market makers' and they run the UK stock markets.
A 'Bull' market is a term used for when the majority of all shares are on the rise and the economy looks as if it is prospering. So, for example the 1990s were a 'Bull Market'. A Bear market meanwhile, is a reference to a time when shares decrease in value consistently. The Wall Street Crash which happened in the 1920s is a good example.