Hi Yogastevi,
You can sign on any number of iTunes accounts on the same computer. Do you have an Apple ID? If so, then sign out your daughter's iTunes account by going to the iTunes Store, click on your daughter's logon id, and then click on Sign Out in the dropdown. Then click on the Sign In option in the same place, and sign in with your Apple ID.
You will probably need to set up a credit card on your Apple ID if you don't have one already set up. Also, any apps, music, etc. purchased with one of the Apple IDs will always be tied to that Apple ID, so if you need to update any of the apps purchased by your daughter that you have downloaded to one of your devices, you will still have to put in her password to do the update.
You also need to make sure you have Authorized the computer with your Apple ID. So, once you have signed on, click on "Store" in the top menu bar, and then click on Authorize this Computer. That will then allow you to play music you have purchased or to re-download anything you have purchased to the Mac.
You both will be able to play music on the Mac regardless of who purchased it if you are using the same iTunes library and if you have both Authorized the computer with your own Apple/iTunes IDs.
Hope this helps. Here is a link to the iTunes support site with a lot of other info you may need:
http://www.apple.com/support/itunes/getstarted/
Cheers,
GB