The Sea Peoples were actually several groups of people, including the Sherden, Peleset, Denyen, Lukka, Shekelesh, and others. Their origins are slightly mysterious, as we don't have a specific place we can tie them to, though most scholars put their origins in the Aegean. They weren't really one solid group of people, but a number of tribes that were "categorized" under the label "Sea Peoples".
One particular group of the Sea Peoples that you might be familiar with are the Philistines of the Old Testament (you know, David and Goliath). These are almost certainly identified as the Peleset listed in Egyptian texts as one of the main groups categorized as Sea Peoples. A lot of this answer will be referring to Philistines, as they are one of the Sea Peoples that we know the most about.
I have spent many years excavating Philistine remains in Israel, and I can tell you that they are clearly a foreign group to the region of modern Israel, and most likely were from the Aegean area, possibly Greece or one of the many islands between Greece and Turkey. We have not been able to get more specific than that. Their pottery style closely resembles Mycenean pottery from one of the major Greek city-states of the time (the earliest of this stuff excavated in Israel is referred to as Mycenean IIIc), and they have other quite Aegean features in their material culture such as square hearths, and their architecture is laid out in quite a foreign fashion. You can see some of their architecture in the Ashkelon 1 volume listed among the sources below. A plan of some buildings featuring these things can be seen on page 263.
They were accomplished seafarers, according to many illustrations of them by the Egyptians and by themselves. However, though they have left symbols, we have not been able to decipher any of their writing, so we do not have their side of the story. At this period of time, the only true, deciphered writing systems in the Ancient Near East were in Egypt (hieroglyphs) and Mesopotamia (cuneiform). Writing was just beginning to catch on in the areas in between. So that's why they did not leave any writing. It's probably not a "true writing system", but rather a symbolic proto-writing system. Either way, we haven't figured it out yet.
As for the famous battle with Egypt, it is memorialized in the funerary reliefs at Medinet Habu, where massive wall murals depict the sea peoples battling the Egyptians. This is as close as we can get to seeing what they looked like. The Philistines (and perhaps other Sea Peoples in general) liked to have upswept spiky hair, which is one of the ways they are always differentiated when illustrated. There's a piece of Philistine pottery which also depicts a Philistine sailor on a ship with similarly spiky hair, but I can't for the life of me find a photo of it.
To be honest with you, while most scholars believe that Egypt took a major hit in that battle, some are now suggesting based on very recent evidence that they had significant influence on early Philistine settlements that the Egyptians won that battle. Egypt was not the same afterwards, it is true, but the Sea Peoples were defeated after ravaging the land from Turkey all the way down the coast.
One popular myth is that the Philistines were, well, philistines. That is, that they were unartistic brutes, opponents of enlightenment. However, this is actually quite far from the truth. They were quite artistic in fact. Their incredibly decorated pottery is some of the most beautiful of the Ancient Near East until millennia later when glazes were invented. They were quite social, they adopted many customs and aspects of local culture in an attempt to fit in, to make a home.
Now, there were other groups besides the Philistines. One prominent group was the Lukka, hailing from southeast-coastal Anatolia. The Hittite king Suppiluliuma II (ruled 1207–1178 BC, during the time of the Sea People invasion of Egypt) notes in an inscription that
"My father, behold, the enemy's ships came (here); my cities(?) were burned, and they did evil things in my country. Does not my father know that all my troops and chariots(?) are in the Hittite country, and all my ships are in the land of Lukka? . . . Thus, the country is abandoned to itself. May my father know it: the seven ships of the enemy that came here inflicted much damage upon us"
(Jean Nougaryol et al. (1968) Ugaritica V: 87-90 no.24)
Therefore, the Lukka lands were adjacent to the sea, probably southeastern Anatolia near the border with Syria, giving them easy access to boats and shipping.
Here are some sources for further reading on the Philistines and the Sea Peoples:
This wikipedia article on the Sea Peoples is pretty well-researched and will give you an overview of everything.
This is the overview site report for the site of Ashkelon, likely the first Philistine settlement and one of their major strongholds in Israel. (PDF Warning)
Betancourt, Philip P. "The Aegean and the origin of the Sea Peoples." The Sea Peoples and their world: a reassessment (2000): 297-303.. I linked to this book earlier, but this is a specific chapter about their origins.
If you have any questions whatsoever, or if I have been unclear in my answer, please let me know and I will do my best to fill in the details.
