A zombie's energy level is its life energy. It can be found as a red bar at its interface when clicked/tapped on, and it is very important. It gradually decreases when the zombie is working (cooking or serving), and it can be lowered when a zombie gets hit during raids. If it gets too low the zombie will start attacking the customers, which will then scare the rest of the customers away and lower your café's rating. Also, the lower the nergy level, the more often a zombie daydreams. The energy level is replenished by resting, so you should always have some of your zombies resting. It is also gained by eating customers (in raids or in your own café) and enemy chefs, although the energy gained is usually very minimal, thus is hardly effective (about 20% of original).
On a raid, a zombie's health is accounted for by its energy level. It is thus a good idea to close your café and have all of your zombies rest for a good while before sending them on a raid for maximum effect. A zombie loses energy if it gets hit by the chef employees or the raid boss in an enemy raid café. (It has a similar effect to HP.) If a zombie loses all of its health (dies), it will then have to reanimate for 8 hours (now 1 hour) before the zombie is able to do anything again. Every zombie's energy will be at 20 after they reanimate, therefore you should let them rest and not overwork them.
Players can spend Toxin to instantly refill a zombie's energy, and to skip a zombie's reanimation process.
Some different types of customers have energy level-related attributes once infected. For example, Teenagers (and their upgrades) have a faster energy loss rate, but a much faster replenishment rate than normal. Construction Workers (along with their upgrades) are the opposite—slow replenishment, but equally slow loss.
A zombie's patience is dependent on its energy. Some zombies are more patient than others and therefore some zombies will wait longer for their energy to drop before attacking. A zombie with low patience (e.g. Punk) will attack a customer at around 20% energy remaining and a zombie with high patience (e.g. Retiree) will wait until their energy is down to around 2%.
http://zombiecafe.wikidot.com/zombies-list (The website is outdated, but it mentions the zombies' patience.)