I found what I think is called a sidebar slider on a website: http://www.bobbyberkhome.com/home (it's the orange 5% off thing on the left)
I would like to either duplicate this or use something somewhat close. I tried viewing the source, and I see that it is javascript. I looked for a few Word Press plugins but it seems that every time I search for the word "slider" it shows actual slideshows. Is this the wrong terminology?
Lastly, can someone tell me how you go about researching and determining what kind a plugin a website uses?
I thought "viewing source" was the best bet but I'm still a little lost.
I would call it a slide tab - here's a link to a blog post for a jQuery plugin that performs a function like that;
http://wpaoli.building58.com/2009/09/jquery-tab-slide-out-plugin/
It's a few years old but it still gets used for this. I'd recommend against doing this to be honest, particularly if you are planning on putting a contact form in it. I did a lot of work for a lead generation agency last year, we built loads of Wordpress sites with a primary goal of taking enquiries via a contact form. It was important to have the form on every page, so these slide tabs were used.
We ended up scrapping the slide tabs - the conversion rate was absolutely hopeless, nobody was filling the forms contained within the slide tabs no matter what we did with them design-wise.
The second part of your question - depends whether you mean Wordpress plugins or jQuery plugins.
You can't browse through the Wordpress plugins that are being used in a site, all you can do is recognise them from the markup they generate and that you get from experience. Your example is unlikely to be a WP plugin though, just an included JS file and the relevant markup in the templates. Those you can see, Javascript works client side, so unless it's all been minified to make things harder you can go through and look at the included Javascript via the developer console in Chrome or Firefox (or even IE, argh!!).
Most of the time there will be comments at the start of the included scripts for the various jQuery plugins explaining what they do and who wrote them