I'm going to first leap to an assumption that you want this field to hold some value FROM the lookup relationship?

If this is the case, then you can add a formula field to the child object to reference values in the object selected in the lookup easily. Just add a formular field and use the "Insert Field" button to pick the one you want.

ie. If you have two objects:

Shelf__c

Shelf_Item__c

and when you populate a lookup from Shelf_Item__c to the Shelf__c it is on... you could add a formula field of type text (which is read only) and put
 

Shelf__r.Colour__c

in there... this would "copy" (visually) the text value of the shelf colour into the Shelf_Item and put it on it's page layouts etc.etc.


IF you just want to populate a field on Shelf__c with any old text - make read only text field on Shelf__c and then add a workflow rule (from Setup -> Create -> workflow & approvals -> workflow rules

Then make a workflow rule on your shelf object and tell it to execute "created, and any time it’s edited to subsequently meet criteria" 

Set the formula to somthing like

AND(
NOT(ISBLANK(Shelf_Lookup__c),
 ISCHANGED(Shelf_Lookup__c) 
)

which means it will fire when you change the lookup field... and then when you have your workflow rule, use a "field update" to set the text field to whatever you want.

You will have to use the field settings to make it read only remember!