After advising several game dev students on their portfolio websites, I realized I was basically giving the same advice and pointing out the same kinds of issues, and maybe I should write about it. So here's some thoughts for students making a portfolio:
First, figure out your audience. Who is your portfolio primarily for? If you want to get a AAA game job, you should try to tailor it to the norms of that industry -- if it's for some sort of school admissions application, then think about what an admissions committee would want to see. A small indie studio will want to see that you're versatile and that you won't need much supervision to solve problems. Each situation will have different expectations for a portfolio.
A portfolio is more curated than a personal site. A personal site can be whatever you want, and represent all your diverse interests and complex personality. In contrast, someone hiring for a programmer job doesn't care whether you play guitar or whether you draw sometimes, and anyway, they have 50 more applicants to look at! Your still life paintings are important to your identity, but probably irrelevant to a gameplay programming position, so you might want to consider keeping your portfolio separate from your personal website.
Next, here's some more specific advice:
SOME MORE SPECIFIC ADVICE
PUT YOUR NAME AND E-MAIL NEAR THE TOP!!! Don't hide it behind a "contact me" link, don't put it at the bottom. Your ideal outcome is that someone will e-mail you with a big delicious job offer... don't make it hard for them to e-mail you. In game art, some people even argue that you should watermark every single image with your e-mail address and website URL, for the same reasons.
Showcase 3-8 projects. You don't want too few projects, otherwise people wonder "where's all their work?" And you might want to avoid too many projects, otherwise people won't know what to focus on, and if you're a student, your work is evolving so fast that you shouldn't show everything anyway.
Don't write too much. Be concise, and describe each project in a few sentences at most. On group projects, outline your role / contribution / responsibilities. Say what you did and describe it in specific and succinct language. This is also good practice for an interview; if someone asks you about your project, you'll need to be able to talk about the actual work you did. If you just mumble that you helped test the game and provided some ideas, then that won't be very convincing.
If it's not obvious from your work, say your desired role at the top. Do you want to be an environment artist? Maybe put "environment artist" at the top. Are you a game designer or level designer or level scripter? Say so! But don't just say "programmer", there are like 50 different programmer specializations in the game industry alone. If you want a primer to different industry job roles, see Liz England's post.
No splash page. No carousel / slideshow. You're not selling a fantasy or brand; you're trying to convince them that you're a real person who can work with real people. Your front page should basically be a list of links to your projects, a flat and simple navigation structure works best. Until you actually have the experience to act like a loud celebrity, then adopt a tone of modesty, honesty, and authenticity instead.
Test on mobile. Lots of people use phones to browse websites. Many people will probably visit your website on their phone. Hopefully it works on their phone!
A note on hosting. If you're at at a university, they probably offer you free webspace with FTP access, for example NYU has a Web Publishing service. GitHub Pages is also a solid free choice, where your Git repo becomes a web server. But if you don't know anything about FTP or HTML or CSS, you'll probably need to buy a plan at an all-in-one service with a website builder system like Cargo Collective, Squarespace, or Weebly.
Example AAA gameplay programmer portfolio: Hugo Peters. Projects all listed on main page (the front page video is a little obnoxious though)... individual pages (example: "Bolt Storm") list specific systems and features, and Hugo explains the technical implementation and reasoning for each module. Short in-editor videos demonstrate workflow and tuning.
Example AAA game design / level design generalist portfolio: David Shaver. Projects all listed on the main page. Individual project pages have explanation of setpieces and gameplay segments, with specific examples of implementation and short videos. (example: "Titanfall 2, mission 3") Visual aids and short text descriptions explain generalist design touches (example: "Titanfall 2 enemy AI and game feel").
Example AAA level design / scripting portfolio: Mike DiRenzo. Projects all listed on main page. For individual project pages (example: "Rise of the Tomb Raider") the tasks and features for individual puzzles were explained and illustrated with short videos... for projects where he built the entire level (example: "Berserk") there's a layout overview image that illustrates critical path with a walkthrough gallery.
Example generalist game developer portfolio: Jonathan Brodsky. Name and role at the top, list of projects with short descriptions and links, very flat and simple with everything on one page. Pay attention to the project descriptions -- Jon always says what it is and what his contributions were.
Example creative technologist portfolio: Ivan Safrin. When Ivan isn't building his own open-source Unity killer, he's making big impressive interactive installations. Notice how it's a simple gray page with links -- he doesn't need flashy photos or videos, because his work will speak for itself. Also note the concise text descriptions on each project page.
Example 2D/3D artist portfolio: maybe just setup an Artstation profile.
Good luck, and try your best! But even if you don't end up working in games, you'll probably enjoy a higher salary and better quality of life by doing something else. Don't stress out too much, take care of yourself, and remember: it'll probably be OK.