Lovage is a perennial. It's hard to find in England, but I've bought whole plants at garden centres, as well as at a market in Wallonia (where it's called livèche). I haven't seen it sold as a herb in shops. It tastes like very strong celery, but is more fragrant.
It should not be confused with a different herb, called ajwain, whose seeds are sometimes sold as lovage in shops which sell indian spices, where you can also buy hing, which is asafoetida, the silphium substitute.
Also following the liquor's name is the album Music to Make Love to Your Old Lady By, by Mike Patton (Faith No More, Mr. Bungle, etc.) under the pseudonym Lovage. It makes reference to the liquor itself in a tongue-in-cheek fake advertisement intermezzo.
