Hello MrsM and welcome to Seven Forums.
Since you didn't give any info about your computer I'm guessing you have a laptop. It's OK to run a laptop with the battery installed and using the AC adapter. Are you saying with the battery installed you can't power up the computer with AC power?
It's also OK to remove the battery and use just AC power. Will the computer start with the battery removed? If yes, it's likely you have a bad battery.