In basic terms, cloud computing is the phrase used to describe different scenarios in which computing resource is delivered as a service over a network connection (usually, this is the internet). Cloud computing is therefore a type of computing that relies on sharing a pool of physical and/or virtual resources, rather than deploying local or personal hardware and software. It is somewhat synonymous with the term 'utility computing' as users are able to tap into a supply of computing resource rather than manage the equipment needed to generate it themselves; much in the same way as a consumer tapping into the national electricity supply, instead of running their own generator.
Private clouds on the other hand use pooled services and infrastructure stored and maintained on a private network - whether physical or virtual - accessible for only one client. The obvious benefits to this are greater levels of security and control. Cost benefits must be sacrificed to some extent though, as the enterprise in question will have to purchase/rent and maintain all the necessary software and hardware.
As with the other cloud computing categories, IaaS refers to the delivery of virtualised computing resource as a service across a network connection. IaaS specifically deals with hardware - or computing infrastructure - delivered as a service. Offerings include virtualised server space, storage space, network connections and IP addresses. The resource is pulled from a pool of servers distributed across data centres under the provider's control, the user is then granted access to this resource in order to build their own IT platforms. IaaS can provide enterprises with great business benefits.
SaaS is arguably the most common of the cloud computing variations; it's the term used to describe a software delivery model in which applications are hosted (usually by a provider) and made available to customers over a network connection. Many people make use of SaaS without realising it as many web applications are delivered in this way; Gmail, Flickr, Twitter and Facebook are all popular examples of SaaS. Enterprise users also frequently make use of SaaS with many popular accounting, invoicing, sales, communications and CRM systems being delivered this way.