I am looking for a solution to load balancing and failover strategy, mainly for big web applications. We have many services to be balanced, such as web, MySQL, and many other HTTP or TCP based services. But I am not sure what their pros and cons are, and which I should choose.
You should use both: HAProxy is great load balancer and LVS is a solution for failover and avoid a Single Point of Failure.
The most important thing that differentiates the two solutions (LVS, HAproxy) is that one is working at layer 4 (LVS) and the other at layer 7 (HAproxy). Note that the layers references are from OSI networking model.
If you understand this, you'll be able to use one in the right place. For example : if you need to balance based solely on number of connections (let's say), the layer 4 load balancer should suffice; on the other hand, if you want to load-balancer based on HTTP response time, you'll need a higher layer kind of LB.
The drawbacks of using a higher level LB is the resource needed (for the same amount of let's say, traffic). The plusses are obvious - think "packet level inspection", "protocol routing", etc - things far more complicated than simple "packet routing".
The last point I want to make is that HAproxy is userspace (think "far more easy to customize/tweak", but slower (performance)), while LVS is in kernel space (think "fast as hell", but rigid as the kernel). Also, don't forget about "upgrading LVS might mean kernel change - ergo, reboot"...
In conclusion, use the right tool for the right job.