Hi glim, I don't believe Magento searches product tags by default. What you can do is create a new product attribute by going to Catalog > Attributes > Manage Attributes in the admin and clicking Add New Attribute. Give your attribute a code, leave the scope as "Store View", and you're better off leaving the "Catalog Input Type" as Text Field (this will allow you to enter 255 characters).
Set "Use In Quick Search" and "Use In Advanced Search" to Yes. Save your attribute, and go to Catalog > Attributes > Manage Attribute Sets. Depending on your product sets, you might have one or many attribute sets - you'll need to add your new attribute to any sets that contain products you'd like to assign tags to.
Click the attribute set and drag your new attribute from the "Unassigned Attribute" column on the right to the bottom of the "General" section of the middle column. Click "Save Attribute Set" and then go and edit one of your products. Find the new attribute field you've added in the General section and enter the tags you'd like to be searched on. Save your product, and all should be good! If the products are still not being returned, you may need to rebuild your product index (System > Index Management)
Hope that helps!