Let's set the laser part of your question aside for a moment. When a wave of any type (electromagnetic, sound, etc) meets a boundary separating two different media the following can happen (typically in various combinations):
Reflection: Some portion, usually not all, of the wave will be reflected from the surface. Metals are highly reflective in the visible portion of the spectrum. Some metals, like gold or copper, have some absorption (see below) in the blue region so that is why they have a reddish/yellowish hue to them.
Transmission: Some portion, but maybe none, will be transmitted through the medium. Hearing sound standing outside a closed room is an example. Visible light transmission through glass is another.
Absorption: Some portion, again usually not all, of the wave will be absorbed by the surface. This will occur through some depth of the material. This is what will produce heating in a material.
E&M waves striking surfaces around us obviously have a broad range of wavelengths - some in radio wave, some in infrared, some in visible. If you are outside, some in UV.
There is much that can be said about all of these, but since this is a limited space, I'll take the liberty of generalizing. Hopefully most folks will not think I am leaving our something critical - but I can be corrected in this in the comments. :)
Radio waves mainly transmit through most non-metallic materials around us (thus you can talk on your cell phone or listen to an old antenna style radio inside your home). Most visible light is reflected within a very short distance of the surface of materials (we see things almost entirely through reflected light).
It is thus mostly infrared radiation that will induce warming of materials. This is because, in some cases, it can directly excite molecular vibrations in materials. The molecules absorb infrared photons and are excited to higher vibration/rotation quantum levels.
But mainly it is because the molecules, or ions that make up the structure of the solid are slightly distorted by the radiation. Because they are charged or may have dipole moments, they can interact with the E&M field. If you make a simple model of an atom as an electron bound through a spring-like constant to the nucleus and write out a simple equation of motion, it will include a damping factor which is responsible for the absorption. E&M waves interacting with a model system like this represent an example of forced oscillations.
In the book Waves by Frank Crawford, appendix 9 is devoted to E&M radiation in material bodies. There he writes:
The damping force represents transfer of energy from the oscillating charge to the medium. This energy is no longer in either the electromagnetic field components of frequency \omega or is it in the oscillation energy of M but is instead in the form of translational and rotational energy of the atoms, and also of "random" vibrations at other frequencies. It is called HEAT.
Much, much more could be written about this subject. At a higher level, Ziman's Principles of the Theory of Solids, discusses much about E&M waves and solids in chapter 8. But this is in essence how E&M waves heat materials: Absorption of the wave, which will occur over different depths depending upon the wavelength of the radiation, will induce motion in the material constituents.
Also, here is a great Physics Today article on atmospheric heating by IR radiation.
There is also a nice general discussion of light and matter by Victor Weisskopf found here.
With specification to lasers and ablation, see this PSE answer. Hope this helps!