BJT transistors AND gate
Your basic circuit is a shot in the right direction BUT it would only work properly if you would use NMOS transistors insteat of bipolar ones.
The problem with NPNs here is that if Q1 is off but you're switching Q2 on, Q2 will behave as a diode pulling the voltage across R3 and D1 up. So D1 starts to light up (a bit). When you also switch on Q1 then more current can flow and Q2 will operate in saturation, the LED will light up more.
If you can, try using NMOSFETs, two 2n7000 would to the trick I guess.
It can also be done with NPNs but then you would need to build a NAND first and then invert it's output OR (even simpler) connect the LED in series with R2 in the schematic from this tutorial.

Place LED in series with R2: Done :-)