LCCDE in simple words?
An $N$th-order linear constant-coefficient difference equation (LCCDE) is of the form

$$y[n]=\sum_{k=0}^{M}b_kx[n-k]-\sum_{k=1}^{N}a_ky[n-k]\tag{1}$$

It is linear because the sequences $x[n]$ and $y[n]$ appear linearly in $(1)$, and it has constant coefficients because the coefficients $a_k$ and $b_k$ do not depend on the index $n$.

LCCDEs are important because they can be used to describe many practically useful discrete-time (or discrete-space) systems, such as linear time-invariant (LTI) filters (which are also called linear shift-invariant (LSI) filters, if $n$ does not represent a time index).

Given appropriate initial conditions, Eq. $(1)$ can be used to recursively compute the sequence $y[n]$ given its past values and the current and past values of the sequence $x[n]$. For causal LTI filters, $y[n]$ is interpreted as the filtered output, and $x[n]$ is the input. In that case, Eq. $(1)$ describes an infinite impulse response (IIR) filter.

If $a_k=0$, $k=1,\ldots,N$, Eq. $(1)$ reduces to

$$y[n]=\sum_{k=0}^{M}b_kx[n-k]\tag{2}$$

which is a convolution sum, and the coefficients $b_k$ represent the system's impulse response. Note that the impulse response has finite length $M+1$, i.e., the system described by $(2)$ is a finite impulse response (FIR) filter.