First of all, I am new to this, but just fooling myself, trying to figure out how to achieve something like this, and get it to work, it's hard to believe the answers here.
We are talking about phpMyAdmin here, which has endless options. You CAN choose to export the origin table with the parameters - renaming the database and the table in progress - telling someone to rename the existing database or table is irresponsible at best. No need for this.
Besides, no one seems to mention that we really want to import COLUMNS into an existing table - the table is the same in all respects, with the possible exception of the table and the database name, right?
Here's what I did: I did a simple export of the origin table, which is identical, with the exception of the name, as the target table. I opened the sql file in SQL Management Studio (without MS, and you can also use something like TextPad to do the same), and deleted all the things at the beginning about creating a database, tables, etc. (Put it where it says "INSERT" - leave it). Then I replaced any instance of the orgin table name with the name of the existing target table. Then you will see all the data columns in this sql file. Delete everything after the last column that you want to import. Nothing more is needed. Save this file with the name of the target table. Import this sql file - nothing is needed. Done.
COMMENT. If you already have existing columns in the target table, you will be mistaken in the first column because you have a primary key that matches the target that it is trying to use in the source columns. Honestly, I do not know how to fix this. I had 7 columns already (primary keys 1-7) in the target table, so I just cut these columns out while editing the sql file inserted at the end of the column list and changed the keys to be sequential from the last. You must also change the previous end of the last line; to, and then a new last line from, to; I hope if you need to do this, then someone with more experience can suggest a better way to do this - for example, ignore imported keys and automatically increase them when importing. Or the best way to export so that this part automatically takes care of the import. As I mentioned earlier, there are TON export options, including the ability to correct database names and tables during export. I am sure there is something about auto-incrementing the primary key.
By the way, when you say "select a table", do you intend to click on it in the list of tables on the left, and then click the import button on the right? I thought it meant something, if that was what you had in mind. This means nothing. The import button imports the sql script - this is it. What happens, where it happens, etc., depends entirely on the script itself.
Hope this helps someone.