We know that the universe is approximately 13.7 billion years old. What methods did we use to discover this? How sure are we about its accuracy?
The correct version of your first sentence would be something like:
Every work quotes slightly different values for the age, depending on methods, observations used, assumptions, etc. As example, the first results from the satellite Planck report an age of 13.813, with an uncertainty of 58 million years (0.5%, to my knowledge one of the best).
So, first question is answered.
About 100 years ago a number of people realized that our Galaxy is not the only one in the Universe and that the all1 other galaxies are receding from us with a speed increasing with distance (credit for this work is generally given to Edwin Hubble for his Redshift-Distance relation for galaxies formulated in 1929). This together with the theoretical results from general relativity, brought scientists to believe that our Universe is expanding. Which means that in the universe's past it was smaller, much smaller than it is now 2.
The age of the Universe depends on the rate of expansion at each time. To have this, you must estimate the mean composition of the universe (in various components like radiation, matter, dark energy, and curvature) and the present day expansion rate (also known as the Hubble Constant). Once you have this, it's easy3 to compute the age of the Universe.