Carry is the return you get for holding on any security. Bonds, futures and even equities have carry.
Bond carry is defined as the current yield you receive in payment from the bond, plus a capital gains component - the price appreciation you get when the bond becomes shorter.
To explain this latter point, shorter bonds have lower interest rates than longer ones (in general but not always).
When time passes a bond “rolls down the curve”: maybe the interest rate for a 5Y bond is 10% but for a 4Y bond it is 7%. Since prices move inversely to yields, as the bond becomes shorter-lived its price rises.
Future carry is the difference between the price of the future and the price of the underlying commodity.
For example if a barrel of Oil costs $110 but the 1Y future for oil trades at $100, then you can buy for $100 something that’s worth $110. That means you have a 10% ($110 / $100 - 1) carry embedded in the future if the price of Oil does not move.
The concept of carry can be applied to a degree also to equities, as they carry dividends.
A share option that is worth $100 and carries a $5 dividend per year has a carry of 5%.
Of course when the dividend goes in payment, the share will go down by a commensurate amount, but the expectation is that the value will be regenerated as the company will be able to pay the dividend again next year.
Currency crosses have carry as well.
Taking the case of EURUSD, holding the cross is like holding a EUR bond and selling a USD bond.
If EUR interest rates are 5% and USD interest rates are 3%, then holding the cross has an intrinsic carry of 2% (= 5% from the long EUR position minus 3% from the short USD position).