Macro's are short for Macronutrients. The three main macro's are Carbs, protein and fat. So macro counting is basically tracking the amount of carbs, protein and fat you eat in each meal and each day. Each macro has a different calorie count, so in turn you will also be calculating your daily calorie intake.
Lets say you want to give counting macros/calories a try. There are two steps to starting. First, you need to figure out how many macros/calories you should be consuming per meal and per day. This is usually done by a professional or an app. You can calculate your own with some research, but is best to consult a professional so they can tailor it to specifically to you.
Once you know what your numbers are, you need to measure out all of your meals. This is why most people take one day to meal prep and to measure and count all of their food. To track this, you will need a food scale to weigh all of your food, then you can plug it into an app that will track and count for you.
Macro counting is not for everyone, it can be very time consuming to start and it makes it really difficult to track if you eat out a lot or maybe your family isn't going to want to eat what you are always eating to hit your macros for the meal.
But if done correctly, this is a great way to get your nutrition under control and to really evaluate and see what you are really eating. It might also be easier for you to take one day to get everything done for the rest of the week, and then all of your meals are already pre made and measured out for you.
If you want to learn more, you can reach out to a nutritionist, or some helpful macro tracking apps are myfitnesspal, RP diet coach, and LooseIt.