Ahh, indirect fire. My favorite topic.
Ok, what has already been stated is fairly accurate. All fire is indirect in the sense that it arcs at least a little bit. There's no way around that unless you're using a laser weapon. Missiles or guided projectiles that glide might technically be an exception as well, but they have to generate lift somehow or they'll arc like anything else.
Indirect fire really means any time the weapon being employed is being aimed by someone other than the operator; an observer of some sort. Indirect fire, in that sense, came about one it was possible to send target information from an observer to a gun quickly enough to make shooting like this practical. Essentially, when the first field telephones appeared.
Indirect fire requires indirect lay
. Basically, what this means is that you establish the gun's position, direction of aim (azimuth) and altitude relative to some known point on earth. The target is then located relative to that same known point. That requires some sort of accurate mapping system including a projection, sphereoid and all kinds of other highly technical details. You establish the known point you're going to fire off of ideally by having it surveyed in. The more accurately the gun is laid and the point is known, the more accurate your fire will be. With very modern systems like Paladin, GPS and inertial navigation can be used as well, but you still want to start from a surveyed point.
Indirect fire is not simply shooting in an arc to bring a projectile down on top of an enemy or because you have to shoot in an arc to get your projectile there. That's direct fire, just a more difficult, complicated version of it. When discussing, say, WWII naval battles, we never say that 2 battleships engaged each other with indirect fire. The 2 ships each directly located, aimed, and fired at their enemy once the engagement actually commenced, using either visual means or radar. Their fire was not generally directed at the enemy ship by an outside observer. On the other hand, an observer might very well direct ships firing at shore targets, making it indirect even though the ballistic performance of the projectile was similar in both situations.