Skill is an objective measure of your ability to do something. How long you can hold your breath is a measure of your skill: the longer you can hold it, the more skilled you are. How fast you can run 100 meters is a measure of skill: the less time it takes, the more skilled you are. Your score on a test is a measure of skill: your skill at the subject and at test taking.
Talent is a measure of your advantages over other people. Some people are born with a bigger lung capacity, have a genetic advantage like the Tibetans do for oxygen metabolism (article here), and so can naturally survive longer on one breath than others. Some people have the right genes to have the right muscles for sprinting vs long distance (article here), and so can accelerate and run faster than others. Everyone sees the world slightly differently due to how the brain is wired and what kind of childhood someone has: if your parents are math teachers who tutor you often at home, you will seem talented at school even though it isn’t necessarily due to being naturally gifted.
Out of talent comes a measure of potential: if you believe in a physical reality governed by the physical laws of science, then you know your abilities are limited, and since everyone has different physical bodies, everyone’s limits are different. Someone with talent might jump to 6th grade english while in 5th grade, but someone without talent, with another year, will reach 6th grade english and be on the same level. What this means is that most of the time, if Bob is better than Abe at piano, it is a temporary fact: if Abe practices and Bob stops practicing, Abe will surpass Bob. The only time this is not true is when both Abe and Bob are approaching their potentials: one of them will reach a lower limit than the other no matter how much time and effort is put in.
Skill and Talent are typically used to refer to abilities that are within the physical world, like most of the examples I have shared thus far: holding your breath, running 100 meters, and playing piano. Smart and Intelligence are more closely associated with abilities that are within the mind: English class and music theory and composition. The two words are used in the same way: Smart is a measure of objective ability, Intelligence is a measure of your advantage over others.
Achieving top levels of mastery is 10,000 hours away according to Malcom Gladwell’s The Outliers so have patience, and focus on things you honestly enjoy doing.
Intelligence is the logic of building logic (see What is Logic and Judgement?), Smart is the logic you presently have.
Intelligence is maximizing future freedom of action – Computer Science answer by Alex Wissner-Gross