While minimum system requirements vary from site to site, they're generally pretty low. Any computer produced in the last five years or so should be more than powerful enough to run the gaming software.
Of more concern is the type of operating system you're running. Windows PCs have the most compatibility with online poker rooms, but there are a lot of poker applications that work just fine on Macs too. Mobile devices are supported extensively particularly at sites that have poker clients that run inside web browsers.