A subcontract is essentially a grant within a grant. An institution agrees to be responsible for some of the grant activities and sends the UW a proposal for their portion of the grant. In turn, the UW sends a proposal to a grant sponsor that includes the subcontract.
First, make sure you want a subcontract (vs. vendor, consultant or UW employee): Is it a subcontract? (PDF)
Second, see Incoming Subcontracts - includes details about budget, grant forms, approval process and submission.
Outgoing subcontracts are formal, cooperative "sponsored programs" relationships with other organizations, generally universities or non-profit research centers, that are included as part of proposals with the UW as the subcontractor.
See Outgoing Subcontracts for more details about budget, grant forms, approval process and submission.
See also, UW Policy on Subcontracts