Let $X\subset\Bbb P^5$ be a complete intersection of a quadratic and a quartic. Then $X$ is defined by two homogeneous polynomials $f_2$ and $f_4$ of degrees two and four respectively. The ring $\Bbb C[x_0,\dotsc,x_5]$ contains $21$ monomials of homogeneous degree two and $126$ monomials of homogeneous degree four. This means that $f_2$ and $f_4$ of of the form \begin{align*} f_2 &= a_1x_0^2+\dotsb+a_{21}x_5^2 & f_4 &= b_1x_0^4+\dotsb+b_{126}x_5^4 \end{align*} The note I'm reading makes the statement:
Varying the coefficients of $f_2$ and $f_4$ generally corresponds to complex structure deformations.
What does this mean?
Changing the coefficients of $f_2$ and $f_4$ would yield a new complete intersection $X^\prime\subset\Bbb P^5$ defined by polynomials \begin{align*} g_2 &= c_1x_0^2+\dotsb+c_{21}x_5^2 & g_4 &= d_1x_0^4+\dotsb+d_{126}x_5^4 \end{align*} Are $X$ and $X^\prime$ somehow related? This is not obvious to me.
Let $X, B$ be connected complex manifolds and $\pi : X \to B$ a proper holomorphic submersion. For each $b \in B$, $X_b = \pi^{-1}(b)$ is a compact complex manifold, so we can regard the map $\pi : X \to B$ as parameterising a family of compact complex manifolds $\{X_b\}_{b\in B}$.
Ehresmann's Theorem states that all the fibres of $\pi$ are diffeomorphic, so instead we can regard the map as parameterising a family of complex structures on a fixed smooth manifold, say $M$.
Two complex structures on $M$ are said to be deformations of one another (or deformation equivalent) if they arise as members of such a family.
In the case you're considering, all smooth complete intersections in $\mathbb{CP}^n$ given by polynomials of degrees $d_1, \dots, d_k$ are diffeomorphic. So the only (potential) difference between the two manifolds you obtain is the complex structure. The statement is that the two complex structures you get, while they may not be the same, they are deformation equivalent.