First, the simple solution but not necessarily the best:
Define a value list for your Company ID field in Employees. Select the "use values from a field" option and specify Company ID from the companies table for the first field. Then select the "also use values" option and specify the Company Name field as field 2. Now, the users can see the company name field in the drop down or pop up menu when they deploy the value list, but the value list enters the ID number. If you specify Pop Up Menu and choose to hide the first field, the users only see the company name.
For a drop down list, similar results can be obtained with a bit of layout sneakiness by hiding the drop down list behind the Company Name field from the Companies table. The name field has to be set in the Inspector's behavior settings to deny access while in browse mode for this to work.
Please note that there are also a number of more sophisticated approaches that can make working with larger lists of values easier than this basic approach. But since they build on this basic set up, it's a good idea to get it working first before trying to implement a more complex alternative.