Fatty acids are taken into the mitochondria to be broken down. This makes sense especially if you consider that the acetyl-CoA generated can be directly used in the citric acid cycle and oxidative phosphorylation immediately afterwards. Note that some beta-oxidation of fatty acids also occurs in the lysosome when the fatty acids chains are too long for the mitochondria. 

The basic pattern of saturated fatty acid catabolism is that the chain is broken down two carbons at a time by release of acetyl-CoA. So, the challenge for the cell is to turn a two-carbon alkyl group at the end of the chain into a thioester (remember, acetyl-CoA is . To do this, the cell first desaturates the chain (removes some hydrogens by oxidation) to form a double bond. Then, water is added across the double bond to form an alcohol. Thinking back to organic chemistry, remember we can oxidize an alcohol to get a carbonyl, which is exactly what the cell does. The result is a ketone which reacts with the thiol of CoA-SH to form the new thioester acetyl-CoA, which is removed from the chain in the process of its formation.

When plasma glucose is low (or when plasma glucose is inaccessible to cells as in diabetes) and glycogen stores have been depleted, the cells resort to oxidation of fatty acids for energy. Because the brain cannot utilize fat for energy, though, and because glucose is an important fuel source for other tissues as well, the liver begins to produce glucose from Krebs cycle intermediates like oxaloacetate. Once these begin to run low, Krebs cycle function slows, and acetyl-CoA from fatty acid oxidation builds up. The body's solution to this problem is to have the liver convert this excess acetyl-CoA into ketone bodies, which can be utilized by the brain and other tissues for energy. 

Lipids tend to be created outside and brought into mitochondria. For example, the endoplasmic reticulum makes phosphatidylcholine and phosphatidylserine, which are then moved to the mitochondrial outer membrane. Chloroplasts, on the other hand, create lipids themselves, such as glycolipids. Both mitochondria and chloroplasts require lipids to function.

Peroxisomes have a wide variety of functions, including both the production and catabolism of hydrogen peroxide (hence "peroxisome"). These organelles also perform the first chemical steps in the synthesis of plasmalogens, which are used to make the myelin sheaths around nerve cells. Peroxisomes break down fatty acids into acetyl-CoA. This process also occurs in mitochondria. The hydrolysis and thus breakdown of peptidoglycans, which are found in bacteria, is performed by the lysozyme enzyme, not within peroxisomes. (It is easy to confuse peroxisomes and lysozymes, because they serve some similar catabolic purposes.)

Fatty acids must be transported into the mitochondrial matrix in order to go through beta oxidation. Before they can move into the matrix, they must be conjugated to carnitine. Only then can the newly conjugated compound (acyl carnitine) be taken into the matrix by a translocase enzyme.

In order to enter into the Krebs Cycle, propionyl-CoA must be converted into a similar molecule, because it can not enter into the citric acid cycle as is. So, it becomes succinyl-CoA via a 3 step process.

Chylomicrons, made up mostly of triglycerides, are the lipoproteins with the least amount of  protein (percentage of protein in the lipoprotein). Following, in the order of increasing protein amount are: VLDLs, IDLs, LDLs and HDLs. 

Lipoprotein lipase is responsible for degrading triglycerides into two fatty acids and a monoacylglycerol molecule. It is attached to the endothelial lumen of the blood vessel. It removes triglycerides from very-low density lipoproteins and chylomicrons in the blood. HMG-CoA reductase and synthase are important enzymes in de novo cholesterol synthesis, but do not cause hypertriglyceridemia (high levels of triglycerides in the blood). Acetyl-CoA carboxylase and fatty acid synthase are part of lipid anabolism, form fatty acids and do not cause hypertriglyceridemia.

Cholesterol and lipids are carried in the blood by lipoproteins. HDL are lipoproteins that remove cholesterol accumulated in blood vessels and take it to the liver for excretion in the bile or further processing by steroidogenic tissues. HDL delivers cholesterol to liver cells through the scavenger receptor SR-B1. HDL can also transfer cholesterol to intermediate-density lipoproteins (IDL) using the cholesterol transfer protein. Incorporation of LDL (low-density lipoproteins), not HDL by macrophages leads to formation of fatty streaks in atherosclerotic plaques. This does not remove cholesterol from periphery, but rather contributes to it.