Q: What exactly does an LNO (Officer) in the Army do?

Its a liaison officer. Essentially its the subject matter expert from one organization bridging the knowledge gaps to another in helping cross coordination. Example, the Air Force provides ALO's to the Army. They help Army commanders translate request for air support in a manner that can be effectively coordinated to the appropriate C/JFACC C2 channels.

Air Force squadrons get Army GLO's who help the air guys understand what your army commanders need. Then there are intra-service LNO's. Like in the Air Force there will be an Air Refueling LNO to coordinate support to other air assets.]

Of course I cant answer if your service has reg or not, since I am not Army.

They do whatever their assigning officer/organization tells them to do for the unit they are assigned to. Lt Col (Join to see) is correct, the LNO is supposed to bring a set of skills to a unit/organization they don't have. Many joint commands will have a sister service LNO. A BDE HQ may have a BN LNO assigned to help coordinate movements/routes. Indirect fires will have an LNO at units that don't have that asset so they can communicate back and forth.
While in SOF, I met FA LNO assigned to Group and Rangers who provided communication/coordination of fire support for missions. It can be a cool job, in the right place and time.

The short answer is that there is no true singular FM or regulation that dictates what an LNO does. This extends all the way back the the Korean Military Advisory Group of the 1950's, the Military Assistance Command Vietnam of the Vietnam War, and a host of other organizations. The purpose of an LNO is to link one organization to another. That means you have to have some degree of cross-cultural competency and build a base of cultural awareness, understanding or expertise that allows you to link your current organization to the one you are tasked to work with. This is all big brain stuff that usually takes someone who is a COL or LTC with a very specialized focus to get into the nuts and bolts of on a doctrinal level. The real question is what type of LNO are you seeking information about?