Count simply counts the number of fields that are not empty. What fields it counts depends on syntax and any relationships you define.
Count ( RelatedTable::Field ) for example, returns a count of all related records where field is not empty. Thus, this is the most typical use of count to count records, with the relationship controlling which group of records are to be counted.
Where are the birthdays you want to count? Do you have a table of records where each of these birthdays is in a separate record? Does your "main page" layout refer to this table or a different table in layout setup?
If your "main page" layout refers to the table where these birthday records are recorded, you may want perform a find to locate those reocords, then use a Count type summary field instead o the count function to count your records.
Thanks Phil. I figured out a way to accomplish what I was looking for by using the following script:
Go to Layout ["Contacts-Main"(Contacts)]
Set Variable [$temp_date_1; Value: GetCurrentDate)]
Set Variable [$temp_date_2; Value: $temp_date_1 + Admin_Options::g_Look_Ahead]
Enter Find Mode 
Set Field [Key_Dates::Birthday; $temp_date_1 & "..." & $temp_date_2]
Perform Find 
Set Variable [$$Birthdays; Value: Get(FoundCount)]
Go to Layout ["Switchboard"(Contacts)]
On the "switchboard" layout, I have a text block with <<$$Birthdays>> to display the number of birthdays in the next 7 (or whatever value the user sets with g_Look_Ahead). Probably not the cleanest way to do things, but I'm learning...slowly :)