Is there any way to copy a column from excel to an attribute table's column in ArcGIS?
If the purpose is to link data found in an attribute table to and excel file where the spatial information isn't necessary it may be best for you to export the attribute table as a .dbf. You can drag and drop the file into an open excel table (2010 maybe earlier), once the file is open in excel you can do all of your joining within excel and not ArcMap. It seems you may be more knowledgeable with the excel format and this may be the best approach.
Within ArcMap you can import excel files as a table using the Excel to Table Tool. After importing the table you can join the information by using the join field tool. You simply select a field that is common to both the original data and the excel file you are trying to join.
You need to understand that ArcMap data is stored like a database instead of like a spreadsheet and so you need to use the tools ArcMap provides to efficiently get data in/out of it.
There is a pretty thorough guide here on how to work with excel data in ArcMap.
The actual function you are likely looking for is Excel to Table
edit: or try converting your excel to CSV and use this function.