For relocation costs, those are deductible from your (Federal) taxes under certain circumstances (something like, if you move more than X miles to be closer to a job). If it is just a bonus that is just taxable income.

In the US, if they gave you $1000 straight-up for moving costs, you'd probably see something like:

  1.  Company gives you $1000.
  2.  $750 hits your bank account after withholding.
  3.  You spend $1000 to move.
  4.  You file taxes, claim $1000 deduction for moving expenses.
  5.  You receive $250 refund.

I've seen two options: relocation bonus (pre-tax) or relocation reimbursement (post-tax, usually up to a certain amount).

A hiring bonus would presumably always be pre-tax, since there's no expenses to reimburse.