When a reader clicks on a link, does she have to buy something on that visit in order for us to receive a commission? Or will a cookie be stored in her browser for a certain time period so that if she purchases something within that period, we'll be credited with the sale?

or 

If a viewer of my blog clicks on a link to a merchant, and it is directed to a page for a specific product I am promoting, will I still get commission on sales for any other products that person purchases if they decide to browse?

The answer to both of these questions is yes.  For most merchants, you receive commissions on any purchase a user makes if they arrived on the merchant's site via your site. The duration of the cookies vary from one day to as many as 60 days.

Keep in mind, that if after clicking on one of your affiliated links, your visitor leaves and then goes to another site with affiliated links (one example is a coupon site) and then clicks on one of those links, that cookie would override yours.  The most recent cookie is the one that counts.