Complete circuits are circuits without breaks in it. You can only run electricity through a complete circuit.
Electricity "ignores" hard-to-cross places like gaps and blown light globes and just "picks the easy path". If the electricity can finish where it started, that's a complete circuit.
If you can draw a line following your circuit diagram which makes a loop (it starts on one side of the battery, and finishes on the other side), that loop is ONE OF the complete circuits (you can have 1 circuit diagram with heaps of loops, and every loop would be a new complete circuit).
Examples in real life:
Get a bunch of wires, a battery, and 2 light globes.
Now... if a battery has a wire going from one terminal (side) to the other, that's a complete circuit (and it will fry or explode your battery, showing you how powerful electricity can be and how easily you can lose an eyeball or a finger - don't do this). So you can imagine that circuit, but it's a very dangerous one to actually make. You need a "load" to slow that electricity down! (A light globe will do.)
So, now for a safer circuit. If you run a wire from a battery to a light globe, and another wire from the light back to the battery, that's a complete circuit too (the light should turn on to show you this). Hang some other random wires off your battery, they won't part of a complete circuit yet. (Now you have a weird electro-spider with a glowing eye, sort of... anyway...)
Choose a wire starting on the + side of your battery, and another wire starting on the - side. Join the loose sides of the wire up to Globe #2. NOW that is a second circuit! Yep, one battery driving 2 complete circuits (looks a bit like a figure-8 huh).
Whatever you do, make sure a light globe or other load is on each circuit. If you rip the light out, that's OK. But if you rip the light out THEN touch the loose ends together you just did something totally stupid and dangerous, called making a SHORT CIRCUIT. This kind of circuit is exactly the same as the first imaginary one, but this time, you should know that your whole circuit (actually pair of circuits) will IGNORE the one with the load on it. So your light would go out, and your battery would fry (again).
So that's complete circuits in a nutshell.