There are two main ways by which we can detect exoplanets, and both provide us with different sets of characteristics about those exoplanets.
First, astronomers can observe the behaviour of a particular star. If the star is seen to 'wobble,' this means it is likely that star has a planet or planets in orbit around it, affecting its own speed and movement. It might sound strange that a star, like our own sun, can move, but the laws of physics state they must if they have objects close by, even if they are planets many hundreds of times smaller than the star. Astronomers can then detect the change in the nature of light we on Earth get from that star, or simply the 'to-and-fro,' movement of the star itself. Further study can then reveal the planet's characteristics.
Over 5000 Possible Exoplanets Have Been Discovered
Second, astronomers can detect the exoplanets themselves passing across the face of their star, as we observe it from Earth. This dims the light we get from that star by a tiny fraction, but this is enough for us to know an exoplanet (or exoplanets) is present. Occasionally, an exoplanet might be discovered by accident, through direct observation of a star that scientists did not suspect had a planetary system. The giant planet Fomalhault B was discovered this way.
What we learn of the characteristics of such exoplanets includes their size (sometimes to within 150 miles), the distance from their sun and how long it takes to orbit their sun, what the exoplanet's atmosphere is made of and whether it is rocky (like Earth or Mars) or gaseous (like Jupiter or Saturn). Here's a video produced by NASA explaining some of their exoplanet detection methods: