the analytic part
If the principal part of a Laurent series is the half that misbehaves, the analytic part is the well-mannered half — the ordinary power series with only non-negative powers that you would have written down anyway if the function had no singularity. It is the tame skeleton around which the singular flesh hangs.
Concretely, the analytic part of sum a_n (z - z_0)^n is the sub-sum over n greater than or equal to 0: a_0 + a_1 (z - z_0) + a_2 (z - z_0)^2 + .... This is an honest Taylor-type series; it converges throughout the full disk |z - z_0| < R and defines a function that is holomorphic everywhere inside, including at z_0 itself. So however badly f behaves at z_0, you can always peel off a perfectly smooth piece: f = (principal part) + (analytic part), and the second piece extends nicely across the singular point.
Why care about it separately? Because it is exactly the part you throw away when you want only the singular behaviour, and exactly the part you keep when, after removing a pole, you want the leftover finite value. For a removable singularity the analytic part IS the whole function — the principal part has vanished — which is why such a function quietly extends to a holomorphic one. The split into analytic plus principal part is the cleanest way to separate 'what is finite here' from 'what blows up here.'
Take f(z) = (sin z)/z^3 = 1/z^2 - 1/6 + z^2/120 - ... near 0. The principal part is 1/z^2; the analytic part is -1/6 + z^2/120 - ..., a smooth power series whose value at z = 0 is -1/6.
Separating the analytic part: it is finite at the singular point and carries the regular Taylor information.
Do not confuse 'analytic part of a Laurent series' with 'an analytic function' in general — here it just means the non-negative-power half of one specific expansion, and like the whole Laurent series it can differ from annulus to annulus.