High level Answer:
Trading Book: All the books held in Capital Markets or Investment Banking Division of a Bank. Instruments will include:Swaps, Stocks, Bonds, etc.
Banking Book: All the books held in Commercial or Retail Banking Division of a Bank. Instruments will include:Loans to corporates, Bank Guarantees, etc.
Market Risk Measure: Sensitivities like delta, gamma, vega, theta, vanna, volga, etc. and Value at Risk (VaR)
Counterparty Credit Risk Measures: PFE,EE,EPE,Credit VaR, PD and LGD.
Market Risk in Credit Risk Measure: CVA.
Liquidity Risk: LCR, NSFR, etc.
Answer to @Matias J.:
Market Risk in Banking Book: Most common is called GAP risk. This is difference in collateral value (Book Value - Value Computed by the Lender). Other types can be different spread risk. This can be defined as risk of movement of the underlying value as perceived by the lender vs market value of the underlying.