When you wire circuits in series all the electrical loads (e.g. bulbs) are arranged in a single loop. There is a single flow of current that goes through all the loads which cannot be interrupted. If a single load is interrupted (e.g. a bulb blows out) everything in the circuit goes out. If you couldn't tell which one blew out (a problem with old style Christmas lights) you'd have to check out each bulb individually until you found one that if replaced allows the whole circuit to light up again. Each time you add a load the amount of current in each one goes down (e.g. each bulb gets dimmer). You can only control the whole set of loads with a switch. Compared to a parallel circuit the load on the battery is less so the battery lasts longer.
In parallel there are several separate current loops connected to a battery. When one load goes down it only affects the load in that loop and leaves the rest unaffected. Modern Christmas lights are constructed in parallel so that only one or at most only a few lights go out if one burns out, not the whole set. Each loop contributes current going in and out of the battery so that with a lot of loops this can add up to a lot of current going in and out. This is a safety issue. One, there is part of the circuit with high current. Second, high current means high heat generated in the wires. The insulation could melt and cause either a short or a fire. (The Mythbusters have a great episode which shows that it is the sparks from a short and not the temperature of the wires that causes most Christmas tree fires.) You can control each loop separately with a switch. The high current also uses up the battery faster because because it has to power each loop separately. Finally, if you add a loop the rest of loops maintain their current (so the lights don't dim when you add another bulb).