There are lots of ways to record students nowadays, but Voki.com in particular works well for students who don't like being filmed, and also has a nice text-to-speech functionality if they want to compare their pronunciation. Read on for more suggestions!
You can record your own voice, or type the script and assign a specific accent for the text-to-speech recognition. I've recently used this to provide the context for a lesson on passive forms. I made a newsreader who gave a report about a zombie attack on the college (...was attacked, were hurt) and used it as a dictogloss for the students - they had to listen while noting down as many words as possible, then they worked together to try to reconstruct the report. If your students have access to computers, they could create their own. It's free, but only for a limited number of recordings - if you want to save more, you will have to pay.
I like anything that's free and doesn't require me to set up an account. This is a simple website where you can record your audio (it saves to the website) and share it, for example by emailing the links or posting them on your virtual classroom (if you have one). It also has a few options for 'morphing' your voice - now I know what a space squirrel sounds like! If you are using a mobile device, you might need to download the app instead, but the website works very well on a computer.
Chatterbox and Chatterpix
These are free apps rather than websites, but my students usually use their mobile devices for this kind of thing anyway. As long as they are allowed to download the app, you should be fine with these. Basically, you add a mouth to a picture and then record your voice, then share the resulting video. Chatterpix has a child friendly version which doesn't allow social media sharing, but these apps aren't just for young learners - in my experience, adults can still find these very entertaining.