For ethernet connections there are 4 ethernet ports built into the router. If you need to add more get an ethernet switch with the desired number of ports. Connect the ethernet cable from your router into the uplink port of your switch and then you will be able to use the ports on the switch just as though they were extra ports on your router.
For Wi-Fi connections some routers will have a pre-set limit of how many wireless devices can be connected at once. Usually, that limit can be modified in the router’s wireless settings. For current and recent Windstream routers, only the Sagem 1704 and 1704N have a max limit of 16 Wi-Fi connections at once. That limit can be changed in the Sagem 1704 or 1704N’s wireless settings.
One caveat of multiple devices being connected at once is it’s harder to keep track of what each device is doing. It only takes one device downloading large amounts of data, or uploading to the cloud to use the majority of available bandwidth and cause slow speeds. Routers will generally let all available bandwidth go to whatever device wants it. Some high-end routers may have options to divide bandwidth between devices, or limit bandwidth going to any device.