The word eutectic applies to alloys or other solutions. Alloys of metals often have properties surprisingly different from the constituent metals, and in the case of Sn-Pb solder, the melting point is lower than that of either tin or lead. The melting point varies, depending on the ratio of tin to lead. Eutectic describes the lowest possible melting point or the Sn-Pb composition that has that melting point.
A low melting point can be advantageous, especially if the materials being soldered have higher melting points. For instance, the non-eutectic solder plating on quad flat pack (QFP)-leads will not melt at the temperature used to melt the eutectic solder paste, so the plating remains on the leads. Soldering processes for surface-mount parts typically subject the entire lead to the soldering temperature, so this is important. In contrast, through-hole leads, which are often solder dipped with eutectic solder, usually undergo a wave soldering process in which only the areas of the leads to be soldered are exposed to the soldering temperature. BGAs essentially have no leads, so it is acceptable for the entire solder ball to melt.
Additionally, eutectic solder is better suited for certain applications because of the way that the solder changes phases between solid and liquid. Non-eutectic solder tends to be pasty while melting (which is why it is suitable for plating fine-pitch leads without causing solder bridging). Eutectic solder liquefies more readily, allowing fast and even wetting of the interfaces to be soldered. This is a key reason why eutectic solder is popular for solder paste and BGA solder balls.