On the exchanges where I traded, there are mainly two types of trading. One takes place on the floor (in the pit, downstairs, open outcry, etc.) and another takes place on the screens (electronic trading, upstairs, etc.).
In the Pit
Let's start with trading "in the pit" (officially known as open outcry). Before computers and the internet, exchanges would have trading floors where registered traders could come and buy and sell contracts. The trading floor was divided into many different sections: there would be specific locations to trade (the trading pit), another section to record and reconcile trades, and yet another to take phone calls from your clients. In the 1990's, these trading floors slowly became obsolete with the advent of computers and electronic markets. There are still small pockets of pit trading around today.
Trading on the floor is its own little world. Imagine standing in a group of people, elbow to elbow. A guy with a piece of paper walks in and asks the crowd "My client is looking to buy 50 oil future contracts! Offers?" This question sparks a frenzy among the traders trying to deliver the best price the customer can buy oil for. "I can sell you 30 at 81!" "I have 50 at 80." "10 at 79!" The customer waits a few seconds for the best price. It seems no one can sell it below 79, so everyone is now calling out sizes at 79: "5 at 79" "10 at 79" "15 at 79" "10 at 79!" Meanwhile everyone has their hand up to signal their sizes and their prices. The guy looks around and points to 2 guys and says "I'll buy 10 from each of you," writes on a piece of paper and walks away. The two guys who just sold contracts are also hurriedly writing down the information so they can hand it to their clerks who will enter it into the system.
On the Screen
Now let's imagine trading on the screen. You are sitting behind multiple computer screens; on one screen, you see all the trades that are going through the market and a list of all the trades you've made today. On another screen you're looking at the parameters you use to control the theoretical (or perceived) value of the instruments you are trading. On the main screen you have a list of all the instruments you are trading with a bid price and a bid size on the left and an offer price and offer size on its right. The numbers are constantly changing as people put in and take out bids and offers. Your electronic eye (or automatic trader) looks out for any opportunity to trade based on your parameters (for instance, if the computer sees a chance to buy anything for 25 cents less than its theoretical value, purchase it and notify the user). You sit behind the computer, looking for any opportunities to make a good trade (much like a trader in the pit would be standing around looking/waiting for a good opportunity to trade). However, unlike trading on the floor, the computer looks for opportunity much faster than a human brain can process all the information, and before you know it, your computer makes a sound and a trade appears on your trade list: you've just made a trade!
Advantages and Disadvantages
Speed. Trading electronically means you can let the computer do the heavy lifting in terms of calculating simple profit and loss to evaluating how much risk is associated with the trade; all the arithmetic you had to complete in your head, the computer can do that in a fraction of a second. On the other hand, given the computer's speed and complexity, mistakes can happen before it can be stopped and mistakes can be often overlooked given the number of things that are happening at any given moment.
Counter Party Evaluation. When you're on the floor, you almost always know who you are doing the trade with. Who just purchased 200 shares of IBM stock? Who did he buy them from? These are all questions that can be answered as long as you are aware of what is going on. And using the information on who bought and who sold, you can make inferences about what trades might happen next. However, on the screen, you are not given that information. All you know is that 200 shares of IBM stock traded at a certain price. You have no idea whether it was Goldman Sachs that purchased it (and therefore might be an indication of a bigger trade coming your way) or if it was an individual investor.
The best analogy I've heard between trading on the floor and on the screen is like playing poker at the casino or over the internet. In the casino, you see who you're playing against. You can see their reactions, you can see if they leave the table and come back to sit at another seat. If you play poker over the internet, you might have more analytical tools at your disposal (e.g. percentage of pre-flop folds, etc.) given the amount of number-crunching your computer can do; however, each player hides behind their username. You can't see their reaction, who they're looking at, or even how much alcohol they've consumed.
Most of the trading activity are slowly migrating to the screens, eliminating the inefficient transactions that happen with yelling and gesturing (that's a whole other topic I'll write about in the future). However, some trades still go through the floor, and it would be irresponsible for a trader to completely ignore one side.
When I was trading, depending on what market you were involved in (equity, foreign exchange, metals, energy, interest rate, etc.), roughly 70% of the trading volume were going through the screen.