Evolutionary change occurs gradually. It occurs over millions of years. It begins with a tiny change, or genetic mutation, that is either benign (doesn't harm it's host) or has an advantage.
For example, when Homo Sapiens left Africa, we were already the Human species. We were all the same as we are now. We were all recognisable as humans physically, and intellectually. However, our skin colour was one of the last things to change. That's why there are some human beings who went north to Europe, and retained pale skin, and some that stayed in Africa, and developed dark skin as a protective mechanism against the heat and UV radiation of that climate. This took countless generations.
The eye in most species has evolved countless times, beginning as a light sensitive patch of skin cells, which then developed into a curved surface (so it could detect direction of light) and then gradually becoming more and more specialised, until we have the variety of eyes found in most species.
Modern day evolution? Well, I'm of European descent. Now, back in the 14th century, the Black Death decimated the European population, leaving only 20% of the population alive. Those that survived had certain genetic mutations which, in their descendents, have imparted a strong immunological resistance to HIV AIDS. If you have two parents who carry this gene and passed it to you, your body will be able to hold out far longer against HIV than someone who doesn't have the gene, or someone who only had one parent with the gene who passed it on. That is one of the reasons why HIV is such a serious problem in Africa and Asia -- they didn't experience the Black Death the way Europe did, because of their geographical location. They missed out on the gene.
Another awesome example of evolution is sickle cell anaemia. People with genetic heritage tracing back to Central Africa sometimes carry a gene for sickle cell anaemia. They have red blood cells that are 'sickle' shaped, and because they're a bit wonky, they don't carry oxygen around the body as well. Now, it's not too bad if you just get it from one parent. But if you get it from both, then you have Sickle Cell Anaemia. But the reason it's so prevalent is that people with these wonky red blood cells have a genetic advantage in areas where they encounter the Malaria pathogen. As far as I know, (I read this a long time ago) Malaria has a hard time trying to establish itself in someone who's red blood cell are sickle shaped. So it imparts a defence against Malaria. At one stage in human history, this mutation probably spread through the population because people with the condition were healthier than those who were vulnerable to Malaria, and could reproduce and care for more children. Therefore, their genetic information survived. They had more descendents.
There are magpies in Northern Australia which have evolved a behavioural change which means they can kill the introduced Cane Toad, which is highly poisonous, and eat the only internal organs which do not carry the deadly toxin. They only eat the organs which won't kill them. They are the descendents of magpies which only ate the organs in Cane Toads which didn't kill them. This COULD be evolution. If an ancestor's behaviour prolonged their life, they were more likely to have more offspring, which would probably inherit the genes that drive that behaviour.
Yes, there are problems with Evolutionary Theory. There were problems when it was first introduced. One of the problems was the incredible shortage of fossils. When Darwin first stumbled upon Evolution, there were hardly any fossils, and testing techniques were pretty pathetic. However, we have pretty much filled every missing link on the Human family tree (they are often dismissed by some creationists because they are normally classified as either human or ape. There is no classification for 'missing link'. However, they all fit and show us the metamorphoses from ape like ancestor to human being. We did not descend from Apes, we just share a common ancestor with apes, just like cats share a common ancestor with lions and tigers.) Every year they find more information that further supports Darwin's Theory. It is a success because Darwin proposed it, based on evidence, and now a century later, there is a vast quantity of evidence that supports it. Archaeopteryx is the missing link between birds and reptiles, and has trace indentations of feathers, plus claws on the ends of the wings. They keep finding stuff like this that supports evolution as a theory.
There is an increase in genetic information. They don't know what many of the genes in a human being actually do. Some of them might just be junk. If you take a fruit fly and make a tiny change to the DNA that controls the development of their antenna, they will grow a leg instead of an antenna. Tinkering like this is helping scientists work out what does what. Many of the g