What you're talking about is generally done with an SCR. You can do this with an NPN and a PNP transistor:
simulate this circuit – Schematic created using CircuitLab
Assume that Q1 and Q2 are both off, and the input is zero. Now apply a voltage to R1 (greater than about 0.6 volts). Q1 will begin to conduct, which will provide base current to Q2, and Q2's collector current will provide base current to Q1, keeping it turned on, regardless of the voltage at R1.
EDITED TO ADD -
As G36 has commented, it may be necessary to add a few resistors for real-life operation.
Equally important, this circuit cannot be used for push-on, push-off operation. In order to turn it off, the current through the load must be interrupted while the input is low. In principle, this can be done by adding another transistor.
simulate this circuit
In this version, with the load ON, if the ON input is low and the OFF input high, Q3 will pull Q2's emitter low and turn it off. When OFF goes low again, the circuit should remain off. Note that in this version, the circuit turns on when the ON input goes high, and remains on until the OFF input goes high and then low again.
Per G36's comment, R4 is probably critical, diverting leakage through Q2 away from Q1's base.