As a guitarist or pianist or any instrumentalist that can create chords, it's fun to make them up or make sense of someone else's weird chord they made up. We want to be able to figure it out, write it down, and make it easy for others to play along with our weird-sounding chord progressions.
Step 1: Identify all the notes
First, figure out all the notes in the chord. Write them down. Try to use all sharps OR all flats to avoid confusion later. Here's a weird chord I just made up:
Not very nice sounding maybe, but it's unique. It evokes... disease and plague! The notes are C F# B D#. Using flats I can also write this as C Gb B Eb
Step 2: Find a triad
Try to make sense of the chord by finding a major, minor, diminished, or augmented triad in the chord. Sometimes this isn't possible because notes are left out, but generally you can find the fragment of one at least.
In my chord, I found two triads. One is B major - B D# F#. The second is C diminished - C Eb Gb
Step 3: Deal with notes outside the triad
If I call this chord a B major, I have an added C (a b9). If I call it a C diminished, I have an added B (a seventh just like a major 7 chord would have).
Step 4: Bequeath a name
I could call this chord several different things:
So which do we pick? I personally would choose B/C, because that's very easy to read at a glance. Generally, it's good to pick the chord that is easiest to read or that is closest to a chord in your key so it "makes sense" in your progression. The bass note can often give you a clue as to what the chord is "supposed" to be.
As you can see, the further from a basic triad we get with adding 6ths, 7ths, 9ths, 11ths, etc., the harder it gets to definitively name a chord. Take this chord for example:
This is both an Am/C AND a C6 chord! So it's fair to say that, in a musical sense, C6 is equal to Am/C. That's called a chord substitution.
And add to this the fact that when you are with a bassist who is playing a C, you don't even need to play the bass note. So in this case it's fair to say that C6 = Am.
There are lots of chord substitutions out there, and jazz players like them because it makes life easier and more creative. But the concept blends into other genres as well. For instance, sometimes if my chord chart says "C" I'll play a G way up on the neck with just the higher strings. That actually makes a C6/9 chord that sounds great. And of course these kinds of "rules" translate to other chords too... E6, for example, can be thought of as C#m/E.