The history of physics is intimately bound up with the history of mathematics, as Sir Issac Newton was both the greatest physicist in history and one of the greatest mathematicians. One of my philosophy profs had the habit of asking every physicist he met who the greatest was, and he said that in twenty years travelling all over the academic landscape, everyone said, "Newton". He'd expected a few "Einsteins" or "Feynmans", but got not a one.
If you ask a mathematician who the greatest mathematician was they'll almost certainly say, "Gauss", or possibly "Euler". Never Newton. In one case a member of a father-and-son team of mathematicians was asked who the greatest father-and-son team of mathematicians was, and he answered, "Gauss... and his father". Gauss' father was a bricklayer, but Gauss was more than enough mathematician for both of them. So while I'd like to claim top spot in two fields for Newton, I'll acquiesce to the judgement of my mathematical colleagues in this matter (if no other).
Newton needed new mathematical language to express the thoughts he was having about motion, and the language he invented was differential calculus, which has been spoken--sometimes more as a pidgin than a genuine language--by physicists ever since. It has also been taken up by many other fields, and one can find differential equations in everything from psychology to evolutionary biology to geology.
Where-ever things are changing, there's a differential equation waiting in the wings, ready to come out and describe the situation.
Given this, talking about science without talking about differential equations is a bit like talking about Formula One without talking about wheels. But unlike wheels, differential equations are obscure. Possibly occult (which in the sciences means "hidden", not "supernatural"... mostly.)
That said, most people have used differential equations without knowing it.
For example, Kingston is 250 km from Toronto and traffic on Highway 401 moves pretty reliably at 100 km/hr. Knowing that, most people can see that it takes about two and a half hours to get from Kingston to Toronto.
Doing that bit of mental arithmetic involves solving a differential equation. Understand that, and everything else is book-keeping.
The reason this example is easy is because the value of the derivative does not change over the time we're interested in. If it did change—if the car moved with variable speed because of that huge traffic jam near Oshawa—then we’d need to break the time up into short intervals where the derivative is approximately constant and add up the results, but the arithmetic is just as simple in each one of those intervals, and the adding up part really is just book-keeping. It's what we have computers for.
The differential equation that describes a car moving along a highway can be written in different ways. x = v*t is the usual way of it, where x is distance, v is velocity, and t is time. The faster you go, the further you travel in a given time.
The equation can also be written in an explicitly differential form:
x = dx/dt * t
Here, dx/dt is the first derivative of position (x) with respect to time, which we call velocity and generally replace with the symbol "v" so people won't realize they're solving a differential equation every time they estimate how long it'll take to get to Toronto.
A differential equation is just an equation that involves the derivatives of some quantity. The "quantity" can be anything from position to the strength of the electro-magnetic field to the pressure in a fluid to the prevalence of some belief in a population. We can have equations that relate the space and time derivatives of complex functions like the quantum mechanical wavefunction to each other, as Schrodinger's equation does, and we can have simple equations like the one above, that relates how fast something's position is changing with time to how far it goes in a given time interval.
Not everything can be described by a differential equation, mind. One of the conditions for such a description to work is that the property in question--position, field strength, pressure, belief, whatever--be more-or-less continuous. This is due to the nature of derivatives themselves.
A derivative is the instantaneous rate of change of one thing with respect to another. In physics the "another" is almost always either space or time, and we are almost always restricted to the first and second derivatives. We even have names for them.
The first derivative with respect to time is called "velocity", and measures how fast something is changing.
The second derivative with respect to time is called "acceleration", and measures how fast the velocity of something is changing.
The first derivative with respect to space is called "slope" and measures how rapidly something changes as we move from point to point.
The second derivative with respect to space is called "curvature" and measures how bent something is. Unlike acceleration, which is a kind of... err... derivative quantity that is best thought about in terms of the rate of change of velocity, it is often more useful to think of curvature as a ding an sich, a thing in its own right.
If a quantity is not continuous--if it can't take on any value within a range but instead jumps in steps, like the number of sheep in a pen--then its derivatives are formally undefined. Fortunately, this rarely matters in practice, because as noted above the real criterion is that the quantity be "more-or-less continuous", and "more-or-less" turns out to cover a wide range of interesting cases.
Consider sheep in a pen: if there are one or two sheep, adding a second or third is a big, sudden, discontinuous change. But if there are a hundred sheep, then adding one more starts to look pretty continuous. Even in the case of a just a few sheep, how fast does the new sheep come in? If it's on the run the derivative may undergo a momentary jump, but in my experience sheep are pretty slow moving most of the time, so there could well be a fairly lengthy period where our pen contains a fractional sheep as the new animal moseys through the gate.
It turns out that this kind of continuum hypothesis can be used in a huge range of circumstances, which allows us to use differential equations to describe all kinds of formally discontinuous systems. Unsurprisingly, these approximate descriptions are incredibly useful and powerful, just like all the other approximate descriptions we use in our daily lives. If I tell someone, "You won't have any trouble finding the place, just take the third left off South Road and keep going 'til you see a house with wooden moose out front" they will not, in fact, have any trouble finding the place, even though my description is barely more than a sketch.
Science is more of an art than a science, and one of the most artful parts is how we apply approximate descriptions to generate useful insights and understandings and predictions. People often have overblown expectations of "scientific certainty" or "scientific precision", whereas in reality a large part of science is managing those things, not eliminating them. There are cases where exactitude matters--mostly when it comes to conservation laws, which are not to be mocked--but in a large number of cases our approximate descriptions are more than good enough for going on with.
So even in cases where we have to approximate like crazy, differential equations turn out to be extremely useful, and because we have a deep toolkit of techniques for solving them and otherwise manipulating them, turning a problem into a differential equation is one of the most powerful tools we have for making reality tractable to our relatively underpowered brains.
Sheep like to flock, for example, so maybe the rate at which sheep go into a pen (dSheep/dt) is related to the number of sheep already in the pen (Sheep) by a simple constant factor K:
dSheep/dt = K*Sheep
Equations of this form, where the first derivative of a value is just equal to the value times a constant, have exponential solutions. This kind of knowledge is part of what makes such descriptions so powerful: there are a relatively small number of physically important differential equations, and by developing a deep familiarity with them we can use our knowledge of the equations and their solutions to reason about reality. What looks confusing complex can often be reduced to a familiar friend.
In this case, since there are a finite number of sheep outside the pen, and this equation tells us that the number of sheep inside will grow exponentially, we can immediately see that we need a better description. It's very nice when you find out right away that your first approximation isn't any good... it saves a lot of barking up the wrong tree.
The thing about all of this is that's its a recent invention, just over three hundred years old. Neither Ancient Greek philosophers nor Medieval logicians had any concept of a derivative. The idea that something could both have a specific value (a car could have a definite position along the highway) and at the same instant be in the process of changing position was incomprehensible to them, to the extent that the Greeks divided their concept of time in two: "synchronic" time described a set of frozen snapshots of reality that came one after another, and "diachronic" time described the process of jumping between those snapshots.
It probably won't come as a great shock to learn that they spent a lot of time being awfully confused about the nature of motion, and struggled with things like Zeno's paradoxes, which resulted from their failure to understand that a thing can both be and be changing at the same time. This is what the concept of a derivative captures: when we say a derivative is an instantaneous rate of change of a value with respect to something else we mean precisely that there is both a definite value and a definite rate of change, all at once, and not a series of discrete values with steps between them.
Once we make that conceptual leap, which Newton and Leibniz both took at around the same time, in the mid-to-late 1600s, we can start to discover relationships between those derivatives, and use those relationships to understand everything from the motion of waves to why there are so many sheep in that pen. Baa.