Degrees, minutes and seconds are all measurements of angles in the sky. Strictly speaking they are degrees, minutes and seconds of arc, to distinguish them from degrees of temperature or minutes and seconds of time. But usually only the minutes and seconds are called arc minutes or arc seconds.
As you learn in school, a full circle – which in our case is the horizon or the distance around a sphere – is divided into 360 degrees, written 360º. So a quarter of that, the distance from the horizon to the overhead point (the zenith), is 90º. Astronomers tend to call the angle of an object above the horizon its altitude, which is a bit confusing, but that’s astronomers for you.
These angles are also used when explaining how big something appears in the sky. The Plough (Big Dipper) is 26º from one side to the other, for example. The Moon is about half a degree across. When it comes to sizes smaller than a degree, we use minutes, with 60 minutes to a degree (written 60′). So the Moon is about 30′ across (it varies a bit over the month).
The planets are a lot smaller, and their sizes are measured in arc seconds (written ″). Jupiter is generally the largest, and varies from about 30″ to 45″. Mars never gets bigger than around 25″, and is usually a lot smaller, while the outer planets are only two or three arc seconds across.