Let $y(x)$ denote a function of $x$. A differential equation is a function that captures the relations between the derivatives of $y$ and the functions of $x$. An ordinary differential equation will contain $y$ as only the function of $x$ (independent variable). An equation containing the partial derivatives will be call a partial differential equation.
\begin{align} F(x, y^{(1)}, y^{(2)}, \ldots, y^{(n)}) = 0 \end{align}
where $y^{(1)}, y^{(2)}, \ldots, y^{(n)}$ denote the various derivates of $y$.
The order of a differential equation is the highest derivative in the equation. For the above case, the order will be $n$ assuming that a term containing $y^{(n)}$ exists in the equation with non-zero coefficient.
Similarly, the degree of a differential equation is the power to which the highest derivative is raised in the equation. Suppose a term $(y^{(n)})^{m}$ existed in the equation where $n$ is the order, then the degree will be $m$.
A differential equation is linear if it can be expressed as follows \begin{align} L(y(x)) &= L(y^{(1)}, y^{(2)}, \ldots, y^{(n)}) = f(x)\newline a_{0}(x)y^{(n)}+ a_{1}(x)y^{(n-1)} + \cdots + a_{n}(x)y &= f(x) \end{align}
where $L$ is a linear transformation from the space of functions that are derivatives, to the space of functions. The coefficients in this linear combination can themselves be functions of $x$, with $a_{0}(x) \neq 0$. Any equation not expressed in this form will be termed non-linear.
A simple example of DE in real life is the motion of a pendumlum modeled as \begin{align} \frac{d^{2}\theta}{dt^{2}} + \frac{g}{l}\sin\theta \end{align}
Suppose we have a differential equation of the form $y^{(1)} = f(x,y)$ and the solution to this equation is $y = g(x) + c$, where $c$ is an arbitrary constant. Then, this solution is called a general solution since it represents a family of curves. The solution can also be of the form $y = cg(x)$, or anything else depending on the differential equation itself.
If we choose a specific value of the constant, we get a particular solution to the problem. The constant can be determined using an initial condition like $y(x_{0}) = y_{0}$. An ODE together with the initial condition is called an initial value problem.