Interactive model of the hyperbolic plane for a general public lecture
By chance I wrote, not long ago, the following applet (HTML5+JS+WebGL) that works at least on Firefox and Chrome.
https://www.math.univ-toulouse.fr/~cheritat/AppletsDivers/Escher/
This work is CC-BY-SA, including the code, but NOT the image by Escher, for which I have not asked permission: you can probably use it a few times in conferences (fair use) but not in a permanent publication (and I will eventually have to remove this image or to create a variant of my own, like Valdimir Bulatov).
Live isometries of a hyperbolic ornament
As part of my dissertation Creating Hyperbolic Ornaments I wrote some Java software which might serve your needs. It's called morenaments conform (as of now). In particular, I can input any Euclidean ornament, hyperbolize that and then perform isometric transformations of the resulting hyperbolic ornament using OpenGL.
If the original ornament is by Escher, then you even get Escher-like artistic content in additon to the hyperbolic symmetry group. Which is something we did for our article Hyperbolization of Euclidean Ornaments. But you have to explicitely ask the Escher Foundation for permission to create derived work of this kind, so it may be easier to start with some other ornament instead. Personally I tend to base a lot of presentations on The Grammar of Ornament.
My software is GPL, so you are free to use it. While I've been using that software for quite some time, it was your question here which made me publish it on GitHub. So it is well-tested on my computer with me using it, but not with other computers or other users. The project README should provide instructions on how to run and use the application for the use case described above. Please report any issues you might have running the code.
Other alternatives
Cinderella does ship support for hyperbolic geometry, in the Beltrami-Klein model or the Poincaré disk model. That way you can create a construction e.g. based on a regular polygon, and then move that around interactively by dragging the center of the polygon. You'd enable hyperbolic geometry using the “Hyp” button at the bottom. Then you see the fundamental circle of the Beltrami-Klein model in the main “Euclidean View” (slight misnomer in this case) and you can open a “Hyperbolic View” from the “Views” menu. It's free but closed source software. I'm currently involved with its development.
iOrnament by Jürgen Richter-Gebert does allow viewing a drawn ornament in hyperbolic geometry. I don't recall whether it was possible to move that ornament. Internally this is built on precompiled lookup images which were in fact created using the software I mentioned up front. This is commercial software for iOS devices only, since I haven't gotten round to writing its Android counterpart yet.
Interactive visualizations in general
Andreas' answer made me aware of the fact that much of the content my former supervisor Jürgen Richter-Gebert created would fit that description of high-quality interactive geometry (and other things) in education. While historically most of that is based on Java Applets exported from Cinderella, the future for math on the web lies with JavaScript, which motivated the project at the center of my current day job: CindyJS. We are still working on a nice gallery for its website, mostly based on this collection of nice interactive widgets. I don't think we have anything particularly hyperbolic at this time, though. The framework is Apache-2-licensed, most content CC-BY-SA.
Not exactly what you ask, but there is a computer game HyperRogue taking place on hyperbolic plane. Having an actual protagonist moving around the plane is an excellent way to introduce hyperbolic geometry to popular audience. Here is an introduction for mathematicians.
The game is free (and open source). Here is an online version that does not require a download.
EDIT: It also supports numerous tilings and quotient spaces now.