Cousin-cousin pairings double the chance of a congenital condition, up from 2% to around 4% -- statistics here. Reportedly, this is about the same bump in congenital conditions as a between a woman having children under the age of 35 vs. over the age of 35 (though the cousins are more likely to have genetic congenital conditions vs. chromosomal errors such as Down's Syndrome). The real issues pop up when you start getting into generation after generation of close breeding/interbreeding, though it's up in the air whether problems arise in the first generation or many generations down the line.

With siblings, the percentage odds would be much higher as statistically the siblings would be sharing 1/2 their genetic makeup vs. 1/8 for first cousins. I'm not sure just how much higher, though. This is, of course, completely ignoring any kind of recessive genetics. From that same article, the risk of inheriting Cystic Fibrosis in the USA at the time was 0.1%. If two cousins who were carriers had children, there'd be a 25% chance of each child inheriting the condition -- a 250x increase.

If one parent is a carrier for a genetic condition, then there ends up being a 25% chance of it being passed down. If it's sex-linked then it differs -- passed down from the male line on the X chromasome, a girl has 50-50 odds of being a carrier, while a boy wouldn't get it at all (as he inherits his lone X chromasome from his mother); passed down from the male line on the Y chromasome, a girl has zero chance while for a boy it's always certain. From the mother's side, it's a 50-50 shot for a boy and 25% odds for the girl. (This is all avoiding such genetic conditions as Kleinfelter's or Turner's Syndrome, which affect the XX/XY sex chromasome makeup and add in varying exceptions.) So, as with Cystic Fibrosis listed above, the genetic conditions become more likely with close relations, either because you already have these recessives floating around, or a mutation gives you a new one which ends up being kept in the family tree.

King Charles II of Spain is a huge historical example of inbreeding -- his family came into power through inheritance rather than war, and so they had the risk that outer-marrying would weaken claims and/or bring others into potentially inheriting the throne. So, they married inwards and kept all the claims in the family. This worked...for a while.

To go back a bit: you inherit 50% of your genes from your mother, and 50% from your father. Philip of Castille and Joanne of Castille provided Charles II with an estimated 40% of his genetic material, despite being born around 200 years before Charles II. His genetic makeup was such that he was more interrelated than someone whose parents were siblings. There were no reported incidents prior to his birth (other than the continuing and increasing size of the Habsburg Jaw -- an example of the lower jaw outgrowing the upper and giving people an underbite and becoming a family trait almost) so for...hmm about 150 years (going from birth to birth) there were no obviously negative effects to all the inbreeding.

Charles II's mother was also his father's niece (practically making him his own cousin); Charles II's maternal (on his mother's side) grandmother was also his paternal (on his father's side) aunt. If you go all the way back the family tree starts nicely with Phillip I's three children all marrying outside of the dynasty...but then one grandchild marries a cousin, whose daughter ends up marrying their uncle; they produce a child who then marries their own cousin. And the end result?

Charles II was born with congenital syphilis. This alone causes severe birth defects and genetic mutations. Beyond that, Charles II was born mentally and physically disabled, as well as disfigured. The Habsburg jaw had either mutated over the years or repeatedly added to with all the inbreeding. It was such that he was reported to have difficulty eating, swallowing, and talking (the latter in part due to an oversized tongue, which also meant he frequently drooled). Did the ruling family realise that continuing the line might be problematic?

Nope. Not at all. Charles II had two wives, both of whom failed to produce children (it was suspected that either Charles II was so genetically malformed that his sperm could not produce a viable embryo, or that his congenital syphilis may have rendered him sterile). Because of this failure to produce an heir, the royal family collapsed and created a succession war. All because one family wanted to try and keep their land and claims to to themselves.