What sound is /a/?

The /a/ phoneme represents the open front unrounded vowel from FATHER and KHAN. If you have this, you do not have the next one. It can be found in some speakers from southern England. Other native speakers use a back vowel here instead, /ɑ/, which means those with the back vowel can easily confuse hearing this one from speakers of southern English for the /æ/ phoneme.

The /ɑ/ phoneme represents the open back unrounded vowel from FATHER and KHAN. It occurs in most native speakers who are not from southern English where they use a front vowel here instead, /a/.

The /æ/ phoneme represents the near-open front unrounded vowel from TRAP and CAT and HAM and MAN. In some speakers from southern England it can be confused for their /a/ phoneme. In some speakers from other parts of the world, including the southern United States and in the Inland North of the Great Lakes of the United States and Canada, it can be confused for their /ɛ/ phoneme, the open-mid front unrounded vowel from DRESS and BET and HEM and MEN.

The /ʌ/ phoneme represents the open-mid central unrounded vowel from STRUT and RUN. It is the stressed version of schwa, /ə/, which is never stressed.

The /ə/ phoneme represents the mid central unrounded vowel schwa from the first syllable of ABOUT and from the last syllable of COMMA. It is the unstressed version of /ʌ/, which is always stressed.

The /ɔ/ phoneme represents the open-mid back rounded vowel from THOUGHT and CALL. Speakers from Western Pennsylvania use this rounded vowel for both COT and CAUGHT, DON and DAWN, while speakers from Southern California and many places west of the Mississippi have no rounded back vowel, so they use only the unrounded back FATHER vowel /ɑ/ for all such words as COT, CAUGHT, CLOTH, DON, DAWN, DOG, GONE — all without any rounding in their mouths. If you are such a speaker then this may be the source of your confusion, as Peter Shor has mentioned in comments.

Here’s a summary so you can see what contrasts with what, and how.

/a/  open       front    unrounded
/æ/  near-open  front    unrounded
/ɛ/  open-mid   front    unrounded
/ʌ/  open-mid   central  unrounded  stressed-only
/ə/  mid        central  unrounded  unstressed-only
/ɔ/  open-mid   back     rounded
/ɑ/  open       back     unrounded