DH-Table*:

α_i-1: Link twist or offset angle (measured from z_i−1 axis to z_i about the x_i axis, again using a right-hand rule. For most commercial manipulators the offset angles are multiples of 90°)
a_i-1: Link length (the shortest distance between z_i−1 and z_i axes. It is measured as the distance along the direction of x_i coordinate frame. For intersecting joint axes the value of a_i is zero. It has no meaning for prismatic joints and is set to zero in this case)
d_i: Link offset (distance from the x_i−1 to the x_i axis measured along the z_i−1 axis. If the joint is prismatic, d_i is the joint variable. In the case of a revolute joint, it is a constant or zero)
θ_i: Joint angle (the angle from the x_i−1 to the x_i axis measured about z_i−1 axis. This is defined using a right-hand rule since both x_i−1 and xi are perpendicular to z_i−1. The direction of rotation is positive if the cross product of x_i−1 and xi defines the z_i−1 axis. θ_i is the joint variable if the joint i is revolute. In the case of a prismatic joint it is a constant or zero)
type: Joint type (revolute, prismatic or fixed)
min & max: min and max joint value respectably In degrees for revolute joints

Although an important convention within the field of robotics, the DH approach to generate the FK for manipulators is far from the only option. Another similar approach is to use screw theory.
If you are familiar with 3D graphics and/or scene-graphs, a serial manipulator is nothing more than a set of parent/child relationships, where the rotation or translation of each child is restricted to zero or one DOF. The kinematics of arbitrary manipulators can easily be setup by combining four different transformation matrices. That is, rotation about X, Y, Z and pure translation. Using this approach you are free to assign frames however you want, and you can define more than just the links and joints contributing to the FK. I.e. defining where hydraulic cylinders are attached to the manipulator. Take a knuckle-boom crane for instance. It is simply R_z(θ₁) * T_z * R_y(θ₂) * T_x * R_y(θ₃) * T_x given a right handed coordinate system with z up.

Inverse kinematics (IK) is normally computed using the Jacobian matrix. Taking the inverse of this matrix you can solve for joint velocities given a desired Cartesian velocity. However, regular inverse only works if the matrix is square. This is not the case for many types of manipulators! Luckily, algorithms exists which lets you solve the IK using the Jacobian even though the matrix is not square! A couple of examples is the pseudo-inverse and transpose methods. However, you'll probably want to stick with the excellent Damped Least Squared (DLS) method, which is the method used by this tool. An excellent paper that covers the three previously mentioned methods can be found here.
Computing the Jacobian for an arbitrary manipulator (or any function) is actually pretty straightforward. To compute the (i, k)-th entry of the Jacobian matrix, simply use the the following formula:

where p_i(x) gives the i-th component of the position function, x_0,k + h is simply x₀ with a small delta added to its k-th component, and h is a reasonable small positive value (say 1E-6).

Here, only the position function is considered. Hence, the IK will only be solved with respect to a desired position. However, the formula can easily be extended to include orientation of the end-effector. Simply increase the number of rows in the Jacobian matrix by three and use the position function for the three first rows, and the rotation (euler angles) for the latter. This is only useful if the manipulator is able to rotate the tool in a sufficient manner though. I.e a 6DOF industrial manipulator.