Anycast is networking technique where the same IP prefix is advertised from multiple locations. The network then decides which location to route a user request to, based on routing protocol costs and possibly the 'health' of the advertising servers.
There are several benefits to anycast. First, in steady state, users of an anycast service (DNS is an excellent example) will always connect to the 'closest' (from a routing protocol perspective) DNS server. This reduces latency, as well as providing a level of load-balancing (assuming that your consumers are evenly distributed around your network).
Another advantage is ease of configuration management. Rather than having to configure different DNS servers depending on where a server/workstation is deployed (Asia, America, Europe), you have one IP address that is configured in every location.
Depending on how anycast is implemented, it can also provide a level of high availability. If the advertisement of the anycast route is conditional on some sort of health check (e.g. a DNS query for a well known domain, in this example), then as soon as a server fails its route can be removed. Once the network reconverges, user requests will be seamlessly forwarded to the next closest instance of DNS, without the need for any manual intervention or reconfiguration.
A final advantage is that of horizontal scaling; if you find that one server is being overly loaded, simply deploy another one in a location that would allow it to take some proportion of the overloaded server's requests. Again, as no client configuration is required, this can be done very quickly.
The short answer is no, unless the application specifically supports it. Otherwise it's up to the IP stack to determine how things get sent out so doing something like putting a weight on one interface is what you have to do.
A workaround for a browser would be to install a local proxy. WebScarab will allow you to do this, if you don't mind installing it on a server. You can set up a local proxy bound to a specific IP and then set your browser to use the local proxy. Thus the request will be sent to the remote server using the IP that the proxy is bound to.
If you use an internal proxy follow this guide to chain WebScarab to that proxy.
Best Answer
This is what I've learned about any-cast...
As far as equipment is required: all you should need is one or more routers that support BGP. Vyatta and PFsense do, if you do not have a compatible router and you feel like going open source. Then you setup each router to broadcast the same IPs.