I want to connect a 10.5v solar cell to a number (say 10 to 20) of 3v LEDs with an ordinary mobile battery for power backup (charging the battery). Should I use a resistor? What else I can do in an effective way?
You need a lot more than just a resistor. You need something to control the current flow through the LEDs. That could be as simple as a bunch of resistors, but a constant-current supply would be more efficient.
You need a proper charging circuit for the battery. Put 10.5V across a normal phone battery, and there's a chance it will catch fire.
A 10.5V solar cell does not really put out that Voltage, it is just capable of that. A solar cell, by its nature, varies with the amount of light, so you need a charging circuit specific for solar panels. A normal cct would use pretty bog standard diodes, whereas I would use schotky diodes, due to the low drop out Voltage. Don't forget you would need a blocking diode in the cct to stop the battery discharging back through the solar panel anyhow.
That is just for starters. Secondly running LED's off a constant current would mean high current drain on the battery. A switched method would be better Pulsed o/p with variable mark space ratio.
Other factors to consider is where you will have the battery/Solar panel. Outside brings its own problems due to heat, both high and low and its affect on charging the battery.