Yes it is possible. You need a strong laser though, with peak powers in the terawatt range.
The key thing for that to actually work over distances longer than a few mm or cm is an effect called filamentation. This is a kind of self-guiding property of air at very high light intensities (starting at TW). There one does not need a collimated beam, just an intense beam and due to nonlinear effects of air a self-focussing takes place which allows propagation of the beam over many km. This effect was not known before one actually had the TW peak power lasers and it took a while to understand the physical reasons behind it.
And yes, this filament in air is ionized and is actually conductive.
There are many ideas how to use this, and one was triggering of lightning in clouds from the ground. While it works in the laboratory, it does not work so well in the field. Check out for example this page (section "lightning application"). I forgot the actual reason, but think it was related to some lifetimes.
Also people think about using it as weapon. Imagine what you can do with an artificially created conductive channel in air that can be km long.