Why your bike needs a battery
Please read the charging system overview first.
While many cars can be driven a long way on a dead battery, you need a functioning battery on your bike to keep riding. Without one it will just up and quit on you. "Newer" motorcycles have solid-state electronic ignition systems, which are somewhat sensitive to variances in supply voltage. In fairly easy-to-understand terms, there are a lot of "troughs" in the AC waveform going into the regulator/rectifier, meaning that the DC coming out of the R/R will not be a nice flat line, but rather a stream of pulses. This must then be "smoothed out", and that is where the battery comes into play. It provides stabilization in the electrical system, as a capacitor would in a linear power supply design.
Note that it is entirely possible to replace the battery in the Ninja 250 with a large capacitor, and that it will run fine this way. 23,000uF is enough (Wes has tested it). The only problem is the starter won't work. It may be something to keep in your emergency kit, though.
Layman's terms
Things that take DC like DC that looks like this:
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - _________________________________________________________(ground)
The 250's DC supply looks more like this, though:
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
_________________________________________________________(ground)
Which really annoys the things that take DC.
But if you run it past a capacitor/battery (they're pretty much the same thing electrically) it comes out like this:
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ __________________________________________________________(ground)
Which is close enough to keep the DC stuff happy.