I bought some round steak to make Hungarian Goulash. I realized I wasn’t going to get to making it in a timely manner and froze the meat. I took it out, defrosted it in the fridge only to have some things come up that made using it right away impossible. Knowing that meat can be refrozen if it is defrosted in the fridge I did just that. However any time I have read about doing this it states that each time the meat is frozen it looses “some quality”. I have not been able to find out what “some quality” actually means.
The general rule is meat should only be refrozen if it still contains a substantial amount of ice/ice crystals. If it is completely defrosted and refrozen, it may be subject to spoilage -- i.e. if it was good for three days before freezing, you freeze it, defrost it and freeze it again it might (or might not) still be edible but definitely won't stand defrosting and holding for any amount of time. The important thing to remember here is that freezing doesn't stop spoilage, it only retards it.
For this reason, I never refreeze completely thawed meat. I use it or toss it. As I have said before on other posts, the best motto is "When in doubt, throw it out!" You don't want to take chances with your health or that of your loved ones.
When meat is frozen, ice crystals that form can puncture the cell walls, allowing liquid to be released. That is why when you defrost meat you will usually find a puddle of watery blood under it. Every time meat is frozen, more damage is done to cell structure which means more liquid loss and tougher meat. And it gets worse each time since the liquid lost on the first thaw causes even more cell damage than before, etc.
Your best bet for meat that has been thawed is to give it a quick fry until it reaches an internal temperature of at least 165°F (74°C). It can then be stored in the refrigerator for a few days before use or colled down and refrozen for up to about a month. While some quality is lost this way too, you have slowed the clock on spoilage.