What are chord inversions?

A lot of the time when we hear a chord, we expect it will be organised with the root note at the bottom - that the order of scale-degree notes, ascending, will be tonic-mediant-dominant (root-third-fifth). If a (triadic) chord is in FIRST inversion, this means we have moved this order downwards by one step, so the THIRD (mediant note) will now be at the bottom of the triad, the root will appear somewhere else higher up, and the new order is (e.g.) third-fifth-root. Similarly, in SECOND inversion, we repeat the same step of shifting the order down by one step - the fifth is in the bass and the order in a triad becomes fifth-root-third (dominant-tonic-mediant). We can even have a THIRD inversion, if the chord has a 7th, where the 7th will be in the bass, the lowest note of the chord. These inversions have a lot of really useful potential roles and functions when writing harmonic chorales - particularly in cadential progressions - so they're worth memorising!