Consistency is a deep topic, and a hard thing to get right. The trouble comes when two nearly-simultaneous changes occur to the same data: conflicting updates can arrive in one order on one server, and in another order on another. This is a problem, since the two servers no longer agree on what the data is, and it isn't clear who is "right".
The short-story: get your favorite RDBMS (for example, mysql is popular) and have your app servers connect to in what is called the three-tier model. Be sure to perform complex updates in transactions, which will provide an acceptable consistency model.
The long-story: The three-tier model serves well for small-to-medium scale web sites/services. You will eventually find that the single database becomes the bottleneck. For services whose read traffic is substantially larger than write traffic, a common optimization is to create a single-master, many-slave database replication arrangement, where all writes go to the single master (required for consistency with non-distributed transactions), but the more-common reads could go to any of the read slaves.
For services with evenly-mixed read/write traffic, you may be better served by dropped some of the conveniences (and accompanying restrictions) that formal SQL provides and instead use of one of the various "nosql" data stores that have recently emerged. Their relative merits and fitness for various problems is a deep topic in itself.