the other party; counterpart; person/company you are dealing with
Primarily used in business contexts to refer to the person or organization on the other side of a transaction, negotiation, or communication.
We sent the contract to the other party.
I will have a meeting with the person in charge at the other company.
取引先 means 'business partner' or 'client' with whom you have ongoing transactions. 相手先 is broader and can refer to any counterpart in a specific interaction, not necessarily a regular business relationship.
Compound of 相手 (あいて, 'partner', 'other person') and 先 (さき, 'destination', 'the other end'). The combination naturally yields 'the other party' in a business or formal context.