TL;DR: The numbers within a chord symbol and their associated alterations (# or b) can be determined by comparing them to the Mixolydian mode.
One point that seems to be missing from the otherwise good answers, when you see most chord notation, you will see that most notes beyond the triad that are specified (2, 4, 6, 7, 9, 11, 13) will be based on the note within a major scale context, so when a flat is added before it, you are lowering it from what would appear in a major scale. The exception is for a dominant 7 chord, where 7 on its own is b7.
Let's look at this in the keys of C major and C minor in terms of scale degrees:
- C: 1
- D: 2/9
- E: 3
- F: 4/11
- G: 5
- A: 6/13
- B: Major 7
All of these notes could be describe as either Major (2, 3, 6, 7) or Perfect (1, 4, 5) but the only one that we need to make this distinction for is the 7, otherwise it is implied. (In case you're not aware, 9, 11, and 13 are essentially synonymous with 2, 4, and 6, though they do function somewhat differently)
- C: 1
- D: 2/9
- Eb: b3
- F: 4/11
- G: 5
- Ab: b6/b13
- Bb: b7
Notice how 2, 4, and 5 are all the same as the parallel major (C major). These intervals are still Major or Perfect, even in a minor key. We would usually make the distinction of the others as being flat because we are comparing them to the major scale.
You can also # chord tones/scale degrees but this is mostly limited to 4 and 5 until you get to more advanced theoretical concepts. So the occurrence of #4 and #5 is relatively common. I'd also note that b4 is not really a thing outside of advanced theory and even there it is still very uncommon.
In the context of speaking about chords, the numbers that we reference are based either on those of the major scale, or the one exception of the 7 being a dominant 7 (b7). I believe the reason that the 7 is the exception here is that earlier music didn't tend to use major 7 or minor 7 chords, primarily just dominant 7 chords (some diminished as well). Since dominant 7 chords came into common use first, as well as still being the most prevalent type of 7 chord, they get the use of the implied meaning.
Context is important here, though. When we speak of a chord and have already declared the type of chord, we usually don't specify the chord tone's quality, or at least we don't need to, since it is implied by the chord name itself. Fro example, on a minor chord, where everyone is aware that I'm speaking about a minor chord, I don't have to say b3, I can just say 3, because all who have studied theory know by definition that a minor chord has a b3.
So what we are trying to understand here, is how and why certain chord tones are specified within the chord symbol. The implication of any number will be either major or perfect, except the 7, unless otherwise stated. So a C chord will be major and we would have to specify that a C minor is a minor chord in the symbol (C min). C7 is dominant but if we want a major 7 chord, we would need to specify that in the symbol (C maj7). For a minor 7 chord (C min7 or C-7), we just need to state that the chord itself is minor (as opposed to specifying that the chord itself is minor and that the 7 is flat) and the understanding is that the 7 will be a flat 7, as that is far and away the most common type of minor 7 chord. Extending this logic to the remaining notes, we need to specify what they are only if they differ from the major scale.
The chord symbol that you've brought, C#min(b6), should be a minor triad with a b6, which would be a minor 6 interval from the root, which is A in this chord. They specify b6 because saying 6 would imply a major 6, which others have mentioned is found in the Dorian mode.
As others have mentioned, this is a very uncommon chord, however, it is possible and you've found it. The purpose of theory is to explain music and provide a language. Depending on what you're looking at, there can be many interpretations and we can't necessarily say one or the other is more correct. Others have mentioned that this could be a first inversion A maj7 chord (A maj7/C#), which would describe the same notes but I can't objectively say which would be a better descriptor without context. If this was one in a series of chords, such as C# min7, C# min6, C# min(b6), then it would make more sense. If it was one in a series of another progression, such as A maj7, C# min(b6), A maj7/E, then it would make more sense to call it A maj7/C#. There are situations where I've seen chords described in ways that seem ridiculous to me, such as G/E, which is really just an E-7 chord, however, that symbol may make more sense to one person than E-7 would and being that they contain the same notes, I can't objectively say that G/E is wrong, though it does not line up with the conventions that musicians use, so I would never choose to describe it as such, other than possibly in a pedagogical setting.
In reality, I could have described all of this in terms of the Mixolydian scale. Mixolydian is the same as the major scale but with a b7, which was the one exception I described. I chose to go from major because of how you framed the question but if you understand the Mixolydian scale, it would be easier to think of it that way. So when you see a chord symbol, you can compare the specified numbers with Mixolydian to determine what notes they should be. Eventually most of this should become obvious and automatic for you, since this is all fairly standard within just about every approach to theory in the western music tradition. Extended chords in a Classical setting are definitely an exception but it's unlikely that most reading this would need to know much about that unless they are in an academic setting, where it should be spelled out fairly clearly for them anyhow.