This type of workout component includes a barbell, your body and a small section of space. Choose about 5 exercises requiring a barbell and arrange them in an order that flows smoothly. For instance, you might follow a front squat with a push press because it transitions well. You probably wouldn't choose a bent over row followed by a reverse lunge since the transition would be awkward. That is an important part of the structure because the tempo is fast without losing form so transitions should be neat and quick. You will complete each exercise for the desired number of reps (say 6) before moving on to the next exercise. You work through all 5 exercises without resting and without putting the bar down. That's one set. Then put the bar down, rest for about 90 seconds and repeat for a total of 4 or 5 sets with 90 seconds rest in between each set.
It will take your breath away and get your heart rate up. Your rest period is your recovery time, so it's like doing intervals. Another important part of barbell complexes is to choose a weight that is manageable for your weakest move. You likely can squat much heavier than you can manage on a row or a good morning. The point is not how much you can lift, the point is shaking your body up metabolically. Choose a weight that allows you to get all the reps in at a fast tempo without putting the bar down. You should need every second of that 90 second rest period to recuperate and you should be working hard enough that you're trying to talk yourself out of the last couple of sets. If you go too light the first time choosing your weight, keep the tempo up and go heavier the next time. Switching weight between exercises defeats the purpose.
If you've never done barbell complexes, give it a try! I'd love to hear about the experience.
If you'd like more information, here's a great article by Alwyn Cosgrove, Complexes for Fat Loss.