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A rep is one unit of a repeated movement, including the return to the start position. For example, one rep on bench press after unracking the weight would be lowering the bar to your chest and pushing it back out to arms length. The number of reps you perform will make up a set. When people write 5x5 they mean 5 reps performed for 5 sets (i.e. a total of 25 reps with breaks in between).
Barbells are the longer bars that people load with weights to perform exercises like bench press, squat, and deadlift. Programmes like Starting Strength are built around these exercises, and I'd highly recommend you pick a programme that includes them if you want to get stronger. Dumbells are the one-handed variation that I think you're probably using at home. They're usually (though not always) of a fixed weight at gyms with the amount written on the side, and are good for things like fixing/preventing strength imbalances between the right and left sides of your body.
Muscle density versus size is typically determined by the amount of weight you perform your reps with, and the number of reps you do per set. It's generally accepted that higher weights for lower reps (1 to 5) will make you stronger and bring about the type of muscle you're after, whereas moderate weights at higher reps (6 to 12) will bring about growth in size, though there is some crossover in effects. The numbers I've put there are pretty widely accepted, but you can search for myofibrillar versus sarcoplasmic hypertrophy if you're really interested in the details.
I'm not sure there's anything I can tell you about cardio beyond that it improves the function of your cardiovascular system (i.e. how efficient your body is at transporting oxygen throughout the body, among other things). It's often employed by people who want to lose fat, those who are conditioning themselves for sports, long-distance or endurance events. Cardio will only help you get stronger if you find yourself out of breath while lifting weights, though there are obvious mutual benefits between being fit and strong.
'Intensity' is a catch-all term for the degree of effort you're putting in during an exercise. For example, in a cardiovascular exercise like running, jogging would be low intensity while sprinting would be high intensity. High intensity exercise will quickly deplete the glycogen (energy stores) in your muscles and you will use up whatever carbohydrates you consume as a form of fast-burning fuel. Low intensity exercise has been shown to use fat stores rather than carbohydrates, but at a far slower rate.
Calisthenics are just bodyweight exercises as I'm sure you know. I'm not an expert on this, but I will say calisthenics are a great measure of relative strength, and are generally a good way of telling whether you're putting on too much fat. Some calisthenics are incorporated into most strength programmes, especially pullups and chinups.
You may find this guide useful for general advice, as far as I remember it's written in laymen's terms:
Exrx is an invaluable resource, particularly in checking whether you're performing an exercise with the correct form, and what part of the body the exercise is intended to strengthen.
This tool will show how strong you are in relation to others:
Read up on programmes, pick one, stick to it. If you're just starting out you can do almost anything and get strong. Keep good form, don't injure yourself, and make sure you're doing this for the long-term. Health and strength are great things to have and maintain over the course of a lifetime, not as a temporary fix to your current insecurities or body image issues. Fit it around your daily activities and be practical about it.