You can make a working target with your Arduino experimenter kit. I see from the pictures that it includes two CdS light-dependent resistors.
Those will change their resistance with light. Hopefully, you have, or will invest in, a volt-ohm meter as it will be helpful.
If you read about what a resistor divider is (see link in my previous post or a bazzilion other links), you can set this up on the breadboard...
The Arduino analog input pin (A0) is capable of reading the voltage at the junction of the divider. It will return a number between 1023 and 0, depending upon the voltage at that pin. The voltage will depend on the resistor values and the resistor value for the CdS LDR will change significantly when it is hit by your laser pointer. The resistance of R1, which is fixed, will not change. So, by monitoring the value of the analog input, you can tell when the target has been hit and when that happens, you can sound a buzzer (you have one of those in the kit as well).
You could also incorporate a potentiometer, which is a variable resistor, in place of, or in addition to, R1 (you have one 10K ohm potentiometer in the kit) - you turn the knob and the resistance changes. This might come in handy when you think about changing ambient light levels.
There is much that I am leaving out, but that is the basic idea, and yes, you could put several CdS cells on the bottom and increase the target area.
So, you have lots of fun experiments to do.
The approach that I am describing is different than using a comparator IC, as described by
rjenkinsgb, but once you set up the scheme above and learn how to do some coding, and exploring, it should help you with using comparators ICs, if you wanted to go in that direction.
Hope this helps.