Convexity, Time and Quanto Adjustments

Under the simplest B-S setting, the price of a forward contract does not depend on the volatility (cf. the price of a vanilla call). This makes sense because while the option payoff is asymmetric, hence would benefit from higher volatility, the forward payoff is symmetric (you lose money when the underlying price goes under water) and hence would neither benefit from nor be harmed by higher vol. However, in some cases the volatility comes into play even when the contract payoff is symmetric, and here we consider a few examples. The common theme of them is that we are trying to price an instrument under the "wrong" numeraire, which introduces an extra drift that does depend on the volatility.

Convexity/Time Adjustment
When the payoff is a linear function of the spot rate. The "wrong" numeraire is the zero coupon bond that does not expire on the right date (i.e. in-arrear swap or in-arrear cap/floor).

Quanto Adjustment
When the payoff is a linear function of the asset denominated in a foreign currency. The "wrong" numeraire is the domestic money-market account.

Reference:
Hull - Options, Futures and Other Derivatives [Chapter 29]
Veronesi - Fixed Income Securities [Section 21.7]
Brigo - Interest Rate Models [Chapter 13]