Help with a program

What I am trying to do is create a laser trip-wire type alarm. when the sensor is triggered, it will log the date and time into memory. I am wondering, will the Arduino store the data, or will I need an external memory device? I am trying to do this as cost-effective as possible, and currently do not have any shields but the proto-shield.

Another option, is to, instead of logging the date and time, set up a seven-segment display to read the number of times the sensor is triggered. Would anyone be able to point me in the right direction with this?

Thank you

It depends how many events you want to log.
The onboard EEPROM has 512 (or 1024) bytes, so storing a Unix ticks style timestamp, you could hold 128 (or 256) events.

Otherwise, I2C EEPROM or SD card.

Awesome, I would only be storing a few timestamps at a time, so the onboard EEPROM would work perfectly. Now, what format is a UNIX ticks style timestamp, and how would I set up my arduino to get it?

Now, what format is a UNIX ticks style timestamp,

It's simply the number of seconds since 1st January 1970, so it's a long (4 bytes). Sadly, it's signed, so it's only good for about 68 years.