No NADH through! Mitochondrial electron shuttles - malate-aspartate & glycerol-3-phosphate shuttles

NADH makes a great electron carrier because it wants electrons, but not desperately, so it’s willing to give and take. You can pass electrons from things with low reduction potential (like NADH) to things with high reduction potential (like O₂), and release little bits of energy as you do, which you can put to use - including to make ATP!  For each NADH that goes into the electron transport chain, ~10 protons get pumped out, and it takes 4 protons coming in for 1 ATP to be made. 10/4= 2.5, so you get ~2 and a half ATP per NADH.  
bit.ly/aspartateshuttle
 
Pretty cool, eh? But that takes place in the mitochondrial matrix, so you have to get the NADH generated in the cytoplasm in there and spoiler alert - you can’t… At least not directly - but you can pass along the electrons its holding onto for you (which is the main thing you care about anyway). And then you can use those electrons to reduce more NAD⁺ inside the matrix so you end up with NADH in there without it ever crossing the inner mitochondrial matrix - sneaky, eh?  
 
In the early sugar breakdown process glycolysis, which takes place in the cytoplasm, you get 2 molecules of NADH (from NAD⁺). But you can’t ship them in because it’s purposefully hard to get into the mitochondrial matrix. You can’t just let any ole molecule in, or else what’s the point of having a compartment anyway, right? So the inner mitochondrial membrane (IMM) is selective about what it lets in (it’s much more selective than the outer mitochondrial membrane so molecules that can pass through generic pores in the outer mitochondrial membrane (OMM) need specialized channels to get through the inner one). But the membrane only has control at the import/export stage - it can choose what to let through but can’t police what the molecules do once they’re in or out…  
 
So your cells find a way to to get those electrons in without taking in NADH itself. Malate dehydrogenase in the cytoplasm transfers electrons generated in the cytoplasm (such as by glycolysis) from NADH to oxaloacetate. Going back to OIL RIG, we can say NADH got oxidized (lost electrons to become NAD⁺) and oxaloacetate got reduced (gained electrons to become malate. This has the effect of transferring 2 electrons (and a H⁺) from NADH (which can’t cross the IMM) to something that can (malate).  
 
So malate enters, and it does so through an antiporter - a type of membrane passageway that does a swap where one thing comes in in exchange for another going out. In this case, malate comes in and alpha-ketoglutarate leaves. So you’ve gotten malate into the mitochondrial matrix - the next goal is to get the electrons it’s transporting to the ATP ATM. And that factory wants those electrons to come from NADH, so now you have to reverse what you had to do to get in. With the help of malate dehydrogenase (the mitochondrial type this time) you take those electrons back out of malate and plop them back onto NAD⁺ (a different copy of it of course, but there’s tons floating around) to give you NADH. This NAD⁺ reduction generates oxaloacetate again. 
 
The NADH is now in the proper location for getting to the ATP factory, so it goes off and gets converted into ATP via oxidative phosphorylation. And now you’re left with the “leftovers” as oxaloacetate. It’s not useful in the matrix, but it can be reused out in the cytosol to transfer electrons again. So you want to ship it out there. But you have a similar problem to the one you had in the beginning. Oxaloacetate can’t get through, so you have to convert it to something that can get through.  
 
This time, instead of converting it to malate, you convert it to aspartate, getting the amino group from the amino acid glutamate. When you take away glutamate’s amino group you get alpha-ketoglutarate (this is the molecule we can exchange malate for in the first swap). This time, in the aspartate swap, we use a different antiporter - the glutamate-aspartate antiporter. It brings in glutamate when you send out aspartate (which is good because we need to replenish the glutamate!) 
 
But the cytoplasm might not want aspartate - and if it does use aspartate for something else, you’re not regenerating the oxaloacetate. So, if you want to keep the cycle going strong you can use cytosolic aspartate aminotransferase to deaminate aspartate back to oxaloacetate, plopping the amino group off onto alpha-ketoglutarate to replenish your glutamate. So basically this malate-aspartate shuttle allows you to interconvert over and over again between the same molecules, with only electrons really getting used up (because they get taken out of the cycle to be used for oxphos). This malate-aspartate shuttle is the main way of moving electrons from NADH into the mitochondria in the liver, heart, & kidneys. Some tissues use other ways like the glycerol 3-phosphate shuttle.
All that still requires oxygen though, so if you don't have oxygen, you can regenerate NAD+ via fermentation (but w/o the energy boost).