Lynn Hill (and several others) have "free climbed" the route.
To free climb a route, you lead it, clipping protection along the way, but climb using only the rock. If you weight any of your protection or the rope in any way, it ruins the free ascent. Once you rely on gear, you are aid climbing.
Aid climbing is when you put your body weight on gear in order to make upward progress. Some "Aid routes" can be done free, usually at a relatively high technical grade. Aid ratings are A0 (bolt ladder) to A5 or C0 to C5. A2 and C2 are similar in difficulty but C means that the gear can be removed while A means that there are fixed points of aid on the route.
IE, Lynn Hill "freed the Nose" in Yosemite at 5.14. But many climbers only free climb up to about 5.11 on the Nose, and then resort to aid for the harder pitches. Most ascents of the Nose can then be called 5.11 C2 (I'm not sure if that is acurate but it's close). The 5.11 is the hardest pitch that was free climbed and the C2 is the grade of the hardest pitch that was aided.
Free Solo means to climb free, without any rope or protection.
Aid solo or Rope solo are two terms used to describe climbing in which the climber is alone, but still uses protection and ropes for safety and/or in order to make upward progress.