The quick search bars on each page only search on the main objects name. If you'd like to search on other fields, or a combination of fields, you can use the Advanced Search and queries at the top of the left hand side-bar.
Advanced Search will allow you to search on any Field in the CRM, but the Queries are the more powerful search. Using a Query you can combine search terms, and use logical operators to combine them in many useful ways.
To reach the queries you can either navigate to 'Campaigns' -> 'View Query' -> 'Custom Query', or you can create your own on the fly from the Advanced Search results page by clicking the 'New Custom Query' button. You can also see previous saves queries by clicking on the 'View/Edit Query' button on the right.
To build a query you add lines of Search terms. First select the field you want to search on, and depending on the field type the operator list will update. Dates will have date operators (such as 'this month', 'this year', etc). Numbers will have Numerical operators such as 'less than', and text fields will have the standard operators such as 'starts with' or 'contains'.
Enter in the correct field, operator and value to match, and click the 'add condition' button. This will add the search term to the query.
Once terms are in the query you have a number of options. You can delete the line, and you can change the Boolean operators. You have the option of adding brackets around terms, so the Query will evaluate these items together as a group (good for combinations of AND/OR searches.
Once you are happy with the Query ensure you have given it a name and save it. The Query will apply to the Search results immediately, and you can view it in the list of existing queries from then on.