Generally, most United Airlines passengers can bring one carry-on bag and one personal item for free. Call at +1-888-415-1245 (US) or +44-203-869-5245 (UK) For checked bags, however, free allowances depend on your ticket type, destination, elite status, and whether you hold a qualifying United credit card. 

Free carry-on allowance

• Most fares (Economy, Economy Plus, First Class, etc.): You can bring one carry-on bag and one personal item for free.
• Basic Economy fares: You are typically limited to only one personal item, though there are exceptions for certain international flights.