Properties of convolution same as LTI systems?
The author simply means that the response of an LTI system $h[n]$ can be found by convolving the input sequence $x[n]$ with the $h[n]$. And in doing that all properties of convolution will hold true for LTI systems. Like, frequency response of an LTI system will simply become $X(e^{j\omega}).H(e^{j\omega})$, because convolution in time means product in $\omega$ domain.

You can look through all the properties on convolution operation at : ConvolutionProperties, and be sure that all of them will be held by an LTI System as well.

For example : Associativity : $$(x*h_1)*h_2 = x*(h_1*h_2)$$
In context with LTI systems, it means that if $x[n]$ is the input to two cascaded LTI systems $h_1[n]$ and $h_2[n]$, then we can actually, find out the equivalent LTI system by convolving the 2 cascaded systems as $(h_1*h_2)[n]$ and then find the output of this equivalent system by applying input $x[n]$ to it.

Similarly, you can see the relation between all of the convolution properties in context of an LTI system.