Hello David,
Thank you for your question.
In case you're talking about PrestaShop multistore database:
http://screencast.com/t/1YAA4LVldRC
you can simply select the products you need, then jump to Shop Associations tab and associate the selected products to the exact store, or store group or even to all stores:
http://screencast.com/t/nZb3XQHyx8H
The same can be done with categories. Select the ones you need then click at one of them with right mouse button and choose Associate with Shop(s):
http://screencast.com/t/kHon7BJSegU
In case you have multiple stores that have different databases, and you switch between the connection:
http://screencast.com/t/0zFSSvAXN then the best way to transfer your products is Export and Import operations.
You can find a detailed instructions on how to export and import products in the articles below:
http://store-manager-for-prestashop-documentation.emagicone.com/prestashop-product-management/export-prestashop-products
http://store-manager-for-prestashop-documentation.emagicone.com/prestashop-product-management/import-prestashop-products