The Finder method is probably the most common method you will come across when using the Business Data Catalog as it is used for the Business Data List Web Part and the entity picker within the BDC Data Column type - two very common uses of the BDC.
The BDC is quite confusing as a method within itself is not a Finder, SpecificFinder or IDEnumerator - it only becomes one of these when it is tagged with the correct MethodInstance name. It has been designed in this way so that through the clever use of parameters and filters a method could be labeled and used as both a Finder and SpecificFinder. Through our experience however it can become very confusing if you do this and tricky to get working - it's a lot lot better to keep 1 method = 1 methodinstance, and this is how BDC Meta Man in fact generates the application definition files. So from now on when I refer to a Finder method, you'll know that I mean a BDC method that has been tagged with the Finder MethodInstance.
Your Business Data Catalog Entity will be a real world object, such as a product. The job of the Finder method is to return the list of products that can be displayed, filtered, sorted - most probably in the BDC Data List Web Part. As most entities map directly to a database table your entity method will quite often do a select against that table, but of course it could do a join across many tables, execute a view, stored procedure - or a web method if you are working with web services.
If you want your users to be able to filter the information being displayed in the BDC Data List Web Part you need to create a filter in your Finder method. How you create a filter depends of if you use a SQL string, Stored Procedure or web method...
1, If you use a SQL String to retrieve your data them you can simply add a WHERE clause with the fieldname and parameter you want to filter.
2, If you are using a stored procedure you need to create an input parameter for your stored proc that will accept it and then do the filtering when it is executed.
3, Web method is similar to stored procedure, you need to create a parameter on your web method that the BDC will pass a value to. It is then up to the code within your web method to do the actual filtering and return filtered results.
So adding a filter to your Finder method adds a number of bits of XML
1, A FilterDescriptor - sets up how the filter text will look in the UI and further UI options
2, A Parameter - the BDC internally uses this to pass your filter value in to be executed
Now you can filter the data in the BDC Data List Web Part

If you are using the Entity Picker with the BDC Column you'll notice there are controls at the top of the form to search the data. The problem is, this won't work unless you have filters defined for your Finder method. It's quite confusing as the controls still appear there if you do not have a filter defined - and people get confused why they are not able to search the data.
Also if you are using the Entity Picker make sure you have the ShowInPicker property defined for the columns you want to appear there, otherwise your users will be very by a list of record ID numbers.