Delta (symbol Δ) means difference. It is always positive, as it is always the greater number minus the lesser number. So the delta of 5 and 3 is 2. Never -2. v is velocity. Velocity is a vector. So yes it is a vector. Δv is the difference in velocity that your rocket or stage can achieve if there is no gravity. Gravity detracts from the potential of the rocket, as does drag. So the rocket spends its Δv fighting the atmosphere and gravity first, and only accelerates after that has been overcome, until orbital velocity is reached. Whatever remains, can be used for missions. Dropping empty stages and boosters reduces me in the equation, thus increasing Δv. That is why staging is so important. Refuelling in orbit increases mf, which also increases Δv.