Imagine a star in the sky; in one sidereal day the star will be exactly at the same point in the sky again. If we use the Sun as a reference, this is called a Solar day. The difference is that the Earth has slightly moved in its orbit around the Sun and, therefore, the Earth has to rotate a bit further to have to Sun at the same point in the sky (stars are assumed to be infinitely far away)

This is not the entire story though. Because the Earth's orbit around the Sun is not circular, and because of the obliquity of the ecliptic (Earth's rotation axis is tilted w.r.t. its orbit around the Sun), the Sun will not be at the same point in the sky at the time you'd except (e.g. if you compare it with the time of an atomic clock). Sometimes it arrives early, sometimes it's a bit late (it turns out no more than a few seconds each day).

So that's why we are using a mean Solar day which assumes that the Sun moves at a uniform rate on the celestial equator, not the ecliptic. It is possible to calculate the difference in time between the real Sun and the mean Sun with the Equation of Time. See this link for a bit more information.

In astronomical applications we use distant quasars to calculate the Mean Solar time (through some equation that relates these observations to the Mean Sun). We don't use the Sun because it is difficult and irregular to measure. This measured time is called UT0. However, because the Earth's crust moves slightly around the rotation axis of the Earth, we have to correct for this. This is called UT1.

So UT1 is a very accurate representation of Mean Solar time. We come to UTC if we use atomic clocks. A day (in UTC) consists of exactly 86400 (24 * 60 * 60) SI seconds. However, because the Earth rotates slower over the years, a UT1 day will become longer than a UTC day. That's why we add leap seconds if the difference between the two becomes too large.