(a) The faster a body is moving the harder it is to bring it to rest, and the more mass a body has the harder it is to bring it to rest (from a given speed). To be more precise, a body of given mass travelling at speed 2v needs the same force applied for twice the time to bring it to rest as the same body moving at speed v. And a body of mass 2m travelling at speed $v$ needs the same force applied for twice the time to bring it to rest as a body of mass m moving at speed $v$. So a body of mass 2m travelling at speed 2v needs the same force applied for 4 times the time as a body of mass m and speed v, and so on. Hence mv seems to be a good way to quantify a body's momentum.
We also assign momentum a direction – that in which the body is moving. This is effected by defining a body's momentum as $m\mathbf v$, in which $\mathbf v$ is the body's velocity. As a vector quantity – see Claudio Saspinski's answer – momentum has remarkable properties.
(b) I expect that Newton started out with the same intuitive understanding of force that we all have. We can feel forces and we can exert forces, perhaps using our hands. Scientists of his era realised that forces could be measured: for example, Hooke's law (1660) makes use of a quantitative concept of force. Newton had the brilliant idea that whenever a body's momentum changed there was always a (net) force acting on it. Even when a ball rolling on a horizontal surface seemed to slow down and stop all by itself, or when a planet in orbit continuously changed its direction of motion. Putting these ideas together (or so we might surmise) Newton proposed that the rate at which a body's momentum changed was simply proportional to the size of the (net) force acting on it. Moreover, the direction of the momentum change was same as the direction in which the force acted. Mathematically, if a constant net force $\mathbf F$ acts on a body of mass $m$, changing its velocity from $\mathbf v_1$ to $\mathbf v_2$ in time $t$, then
$$\mathbf F \propto \frac{m\mathbf v_2 - m\mathbf v_1}t$$
We can factor out $m$ and write this as
$$\mathbf F \propto m\frac{\mathbf v_2 - \mathbf v_1}t\ \ \ \ \ \ \text{that is}\ \ \ \ \ \ \mathbf F \propto m\mathbf a$$
In the SI the units of force are chosen so that the proportionality becomes an equality!
[The force need not be constant; this restriction was made in the argument above just so that calculus notation could be avoided]