Here's a simple series of videos explaining sig figs from Khan Academy on YouTube: Significant Figures (5min), More on Significant Figures (6min), Multiplying and Dividing with Significant Figures (9min), Addition and Subtraction with Significant Figures (9min)
Here's a fun video version of this topic and dimensional analysis (12min): CrashCourse Chemistry: Unit Conversion and Significant Figures on YouTube
Want to learn this material in a more interactive way? Check out my new project, Chemiatria!
Significant means important or relevant. (Insignificant figures are not useful to mention.) Figures can mean a lot of things, but here means digits. Digits are the symbols (1234567890) that you use to write numbers.
Significant figures are the number of digits given in a number. For instance, 18 has 2 sig figs, and 3.456 has 4 sig figs. Sometimes the number of sig figs is a little more complicated: 10 has only 1 sig fig, and 1000 also has only 1. The reason is because you don't have a choice about whether to include those zeros: they have to be there to show what the number is. What about 1001? It has 4 sig figs. We could have "rounded" it to 1000, showing that the last digit wasn't significant, but we didn't. That means that the 1 on the right is significant, and if the smallest (representing 1s, not 10s, 100s, etc) is significant, then the bigger ones (representing 10s and 100s) must be also.
Significant figures have an important meaning. When you measure a quantity, if you report it with the correct number of significant figures, then a person reading the number will know how precise the measurement was.
For example, suppose you measure the length of a box with a normal ruler (price of ruler: ~2000won) marked with mm. You can be sure that your measurement is no more than 1mm different from the real length of the box if you measure carefully. So for instance, you would report the length as 31mm or 3.1cm. You wouldn't have to round it to 3cm or 30mm, because you know more precisely than that. Your reader will interpret 3.1cm as meaning 31mm +/- 1mm. You don't mean 2-4cm, you mean 3.0-3.2cm, so you write 3.1cm, not 3cm.
Instead, suppose you want to know the length of the box much more precisely. Now you will need a better tool. For instance, you could use a dial caliper (price: 35000 or more) to measure to the 0.02mm. Now you could report your length as, say, 31.14mm, meaning 31.12-31.16mm. If you measured with a ruler but wrote 31.1mm, or 31.12 mm, people would probably think that you used a better tool than you actually did, so that would be almost dishonest.
Hopefully all this makes sense. However, it gets a little more confusing if you do calculations. Suppose you were actually measuring the diameter of a circular box, and you need to report the circumference. You measure with the ruler and get 31mm. You do the calculation on your calculator, and get 97.389372261284mm. What do you report? You certainly don't know the circumference more precisely than you knew the diameter. You knew the diameter was in between 30mm and 32mm. So if you do the calculation with both of these, you get that the circumference is between 94.25mm and 100.53mm. So a good rule is if you do multiplication with a measured number, then report the answer with the same number of sig figs as the number you started with: 97mm, 2 sig figs. In this case, it actually means something like 97 +/- 3 mm. The uncertainty is a little bigger than it was before, but you shouldn't write 100mm (1 sig fig) because you don't mean 0-200mm, and you also don't mean 90-110mm. (If you wanted to say 100mm with 2 sig figs, you would have to write 10.cm, or use scientific notation: 1.0 x 102 mm.)
You can look up a lot of rules for how to use sig figs in the textbook, but I would rather that you think about what they mean and just be sensible. For example, I might give you a problem in which 20mL of liquid are used. Technically, if I mean 20 with 2 sig figs, I should write "20.", where the decimal point indicates that the zero is significant. However, I might forget to include the decimal point, particularly in my lab notebook when I'm working in the lab. However, because the standard measuring tool in a lab for volume is the graduated cylinder, I can assume that that's what I used. The graduated cylinder can measure to ~10% accuracy. In this case, 10% is 2mL, so if I used a graduated cylinder, I mean 20 +/- 2 mL, or 18-22mL. Thus, 2 sig figs is correct. In this kind of situation, you can often assume 2 sig figs when only 1 is given officially. Generally, including an extra sig fig, especially in the middle of calculations, is reasonable. In a real situation where someone might make decisions based on your number, you should be careful to use the most correct number of sig figs. For classes, I think including one extra is reasonable, but don't include more extras, because then it starts to look silly.
Some numbers were counted or defined, not measured. These are exact numbers. For instance, there are exactly 1000 g in 1 kg, because that's the definition. Or if you use a volumetric pipette to add 1.00mL of liquid twice, then the total amount added was 2 x 1.00mL = 2.00mL. You used the pipette exactly twice, so the 2 is exact, and you don't have to round to 1 sig fig (2mL) for the total volume.