Ahh, the iconic staple of Paris, the Eiffel Tower! The Eiffel Tower was completed in 1889, a vision of Gustav Eiffel. It was built for the 100th year anniversary of the fall of Bastille, which led to the French Revolution, but also to have a centerpiece for Paris when the World Fair rolled into town a couple of months after completion. At the time, it was the tallest tower in the world and it kept that title until 1930, when the Chrysler Building surpassed it.
Surprisingly, it also used to be the world's largest billboard, when colored bulbs were attached down the tower to advertise for a French car company. It was so bright that Charles Lindbergh used it as a beacon for landing in Paris after his 1927 trans-Atlantic solo flight! (I can't think about Lindbergh anymore without thinking about Philip Roth's The Plot Against America...).
At the top of the tower, you can see Eiffel's hidden away office, where he would spend days working on architecture, talking to his daughter, and meeting with other creatives & thinkers of the time (like Thomas Edison!).
Picnic at the Tower
If you don't have time to go up the Eiffel Tower, or didn't buy your tickets in advance, my number one thing to do, regardless, is to have a picnic in a park under the tower! Most people seem to go to the Champs-de-Mars to get a view of the sunset, but I prefer sitting on the other side in the Place du Trocadero. There are less people, it's cleaner, and you still have a great view! Bring your food and drinks in advance. Don't forget cups and a bottle opener, if you need it!
Visiting the Tower
If you decide to go up the tower, which I highly recommend -- you are in Paris! -- you need to buy your tickets about a month out. Online you can buy elevator tickets all the way to the top floor and you can buy dining, if you really want to splurge! Another option, if you aren't sure what day you can go or if you don't want to commit, is to go there when you want to visit and wait in line for walking tickets. The line can be anywhere from 20 minutes to an hour, but I have never personally waited in line for walking for more than an hour. With your walking ticket, you can walk up to the second level. At the second level you already get a great view of the city, and of the sunset if you went at that time! On the second floor, you can usually purchase elevator tickets to the top, without making the reservation in advance. Go up to the top floor, have your overpriced glass of champagne, and mosey around until the lights start twinkling!