A Steven says, the solution is usually termed a "monostable multivibrator" or just monostable in many cases. There are many means of implementation but a quick cheap easy (chose any 3) way that will probably meet you need is:
Assumes: Applied 0/1 on Voltage shown in diagram is suited to operating a relay AND that it is also accessible as a permanent DC level.
 Relay only solution - no electronics.
Electrolytic capacitor large enough that when charged it is able to operate a relay for long enough to provide required output pulse,
Power source suited to operating relays = Vcharge.
Relay A has a "changeover contact".
Relay B has a single "normally open" contact.
Relay A changeover contact is connected so as to connect the capacitor to Vcharge when un-operated and to swap capacitor onto coil of relay B when operated.
System idle. Capacitor is charged via Relay A contact
Input signal occurs. Relay A operates. Contact on Relay A connects capacitor to coil of Relay B.
Relay B operates, powered by capacitor. Relay B contacts activate output circuit. Relay B remains operated until Vcapacitor falls to Relay B dropout voltage. Relay B then releases
A operated, B released. Stable power on state.
Power to A is removed. A releases. capacitor recharges for next activation.
 One relay plus low cost transistor
Instead of driving Relay B directly, contact A connects a charged capacitor to the base of a transistor via a suitable resistor. Transistor (MOSFET of biploar) is used to operated relay B. Operation is as before except the small magnitude of the base or gate drive is well below Irelay so longer delays or smaller capacitors may be achieved.
 Superb - one relay, low cost, "Friend for life".
A package of 6 x CMOS Schmitt inverters will allow this to be done with 1 inverter (1/6th of package)(Maybe 2 if you want extra features.
Notionally the same as solution  but far more flexible and powerful.
This needs only ONE relay. Relay B is used (if desired) and relay A is replaced with a gate based monostable. Very cheap and easy. You'll probably need a single transistor to drive the relay (or an extremely sensitive relay). Circuit components apart for relay is < $US1.
You can buy a formal monostable IC such as the 74C221  BUT once you learn to use CMOS Schmitt inverters you will have a friend for life. A pkg of 6 such inverters is possibly the most useful analog timing IC available.
Useful for clocks, delays, monostables (retriggerable and one shot), PWM modulators, tone generators, simple switching regulators (I have one in commercial production) ... .
All in a 50 cent (in 1's ) IC . That datasheet is for a CD40106 but there are various alternatives, some about the same in charactistics, some somewhat different. (74xx14, CD40106, xx4584, 74xx19, MC14584, ...)
I can provide a complete (and simple) circuit with values etc IF there is interest. No point in expanding the answer if not of value to people.