Applications and software update keep quitting unexpectedly

First, ensure you have a good external backup. Get a copy of your data, on the off chance there's something wrong.

Then boot from the Recovery Partition or boot from the DVD (depending on your release) and use Disk Utility to check your disk structures and check your permissions, as a start.

You can also launch Applications > Utilities > Console.app and see if there are relevent error messages, though the messages there can be cryptic.

Some information on the OS X version and the type of Mac could be useful in assisting you, as some of the sequences vary by OS X version. If you're not sure, you can use  > About This Mac to get some details.