My husband and I each have an iPad 2. We each have a different Apple ID. Is there a way to merge our IDs so we can share the apps we each have?

Welcome to the Apple Community.

To begin with, I would like to try to explain Apple's confusing nomenclature in regard to iCloud.

Apple have called the whole cloud thing iCloud, there are a number of features under the iCloud umbrella, some of which require their own login. iTunes is one of these, another is what Apple have unfortunately also called iCloud too.

You can use the same Apple ID (account) to login to both iCloud and iTunes, but you don't need to and often users will login to each service using a different ID.

The part that you need to remember is that the services available when you log into iCloud are completely different and unrelated to those when you log in to iTunes. Your iCloud login enables mail, contacts, calendars, find my phone, Back to My Mac, Documents & Data sharing and photosstream, it does not affect any of your iTunes services.

To avoid confusion when discussing your problem, when I mention iCloud, I am referring to the services under the iCloud login, Whereas I will refer to the whole cloud thing as The Cloud.

Unfortunately you cannot merge accounts or transfer content from one to another, you will need to continue to use your own ID's with the iTunes store to be able to continue to use your existing content. However if you were to both use the same ID for iTunes going forward, you can both share any new purchases going forward.