Production is one of the most important game concepts in Civilization V. Producing stuff (a.k.a. "building," "training" or "constructing") means the ability of cities to generate gameplay units or stationary improvements such as buildings or Projects. In this sense, the term "production" is used to express how much a city is able to build each turn - be it constructing a building, or training a unit, or working on a particular project. Since all these are basic activities in the game, and start from its beginning, you can see that it's of the utmost importance to understand and develop production in your cities.
Note that every city has at least 1 Production point from its main tile, so you can always produce something in it. However, if you only rely on that lone point, your projects in this city will take forever. Make sure you expand your Production capabilities if you want your city to progress.