I have given talks about mathematics to non-mathematicians, for example to a bunch of marketing people. Supplemental: to see an example of a talk of mine that was given to a general audience, see my TEDx talk "Zeroes" (with supplemental material). The talk lasted 15 minutes and it took me about two weeks to prepare.
In my experience the following points are worth noting:
- If the audience does not understand you it is all in vain.
- You should interact with your audience. Ask them questions, talk to them. A lecture is a boring thing.
- Pick one thing and explain it well. The audience will understand that in 10 minutes you cannot explain all of math. The audience will not like you if you rush through a number of things and you don't explain any one of them well. So an introductory sentence of the form "Math is a vast area with many uses, but in these 10 minutes let me show you just one cool idea that mathematicians have come up." is perfectly ok.
- A proof of something that seems obvious does not appeal to people. For example, the proof of Kepler's conjecture about sphere packing is a bad example because most people won't see what the fuss is all about. So Kepler's conjecture would be a bad example.
- You are not talking to mathematicians. You are not allowed to have definitions, theorems or proofs. You are not allowed to compute anything.
- Pictures are your friend. Use lots of pictures whenever possible.
- You need not talk about your own work, but pick something you know well.
- Do not pick examples that always appear in popular science (Fermat's Last Theorem, the Kepler conjecture, the bridges of Koenigsberg, any of the 1 million dollar problems). Pick something interesting but not widely known.
Here are some ideas I used in the past. I started with a story or an intriguing idea, and ended by explaining which branch of mathematics deals with such ideas. Do not start by saying things like "an important branch of mathematics is geometry, let me show you why". Geometry is obviously not important since all of mathematics has zero importance for your audience. But they like cool ideas. So let them know that math is about cool ideas.
To explain what topology and modern geometry are about, you can talk about the Lebesgue covering dimension. Our universe is three-dimensional. But how can we find this out? Suppose you wake up in the morning and say "what's the dimension of the universe today?" You walk into your bathroom and look at the tiles. There is a point where three of them meet and you say to yourself "yup, the universe is still three-dimensional". Find some tiles in the classroom and show people how always at least three of them meet. Talk about how four of them could also meet, but at least three of them will always meet in a point. In a different universe, say in a plane, the tiles would really be segments and so only two of them would meet. Draw this on a board. Show slides of honeycombs in which three honeycomb cells meet. Show roof tilings in which thee tiles meet, etc. Ask the audience to imagine what happens in four dimensions: what do floor tiles in a bathroom look like there? They must be like our bricks. What is a chunk of space for us is just a wall for them. So if we have a big pile of bricks stacked together, how many will meet at a point? At least four (this will require some help from you)!
To explain knot theory, start by stating that we live in a three-dimensional space because otherwise we could not tie the shoelaces. It is a theorem of topology that knots only exist in three dimensions. You proceed as follows. First you explain that in one or two dimensions you can't make a knot because the shoelace can't cross itself. It can only be a circle. In three dimensions you can have a knot, obviously. In four dimensions every knot can be untied as follows. Imagine the that the fourth dimension is the color of the rope. If two points of the rope are of different color they can cross each other. That is not cheating because in the fourth dimension (color) they're different. So take a knot and color it with the colors of the rainbow so that each point is a different color. Now you can untie the knot simply by pulling it apart in any which way. Crossing points will always be of different colors. Show pictures of knots. Show pictures of knots colored in the color of the rainbow.
Explain infinity in terms of ordinal numbers (cardinals are no good for explaining infinity because people can't imagine $\aleph_1$ and $2^{\aleph_0}$). An ordinal number is like a queue of people who are waiting at a counter (pick an example that everyone hates, in Slovenia this might be a long queue at the local state office). A really, really long queue contains infinitely many people. We can imagine that an infinite queue 1, 2, 3, 4, ... is processed only after the world ends. Discuss the following question: suppose there are already infinitely many people waiting and one more person arrives. Is the queue longer? Some will say yes, some will say no. Then say that an infinite row of the form 1, 2, 3, 4, ... with one extra person at the end is like waiting until the end of the world, and then one more day after that. Now more people will agree that the extra person really does make the queue longer. At this point you can introduce $\omega$ as an ordinal and say that $\omega + 1$ is larger than $\omega$. Invite the audience to invent longer queues. As they do, write down the corresponding ordinals. They will invent $\omega + n$, possibly $\omega + \omega$. Someone will invent $\omega + \omega + \omega + \ldots$, you say this is a bit imprecise and suggest that we write $\omega \cdot \omega$ instead. You are at $\omega^2$. Go on as far as your audience can take it (usually somewhere below $\epsilon_0$). Pictures: embed countable ordinals on the real line to show infinite queues of infinite queues of infinite queues...
From my perspective, the critical question isn't what must be included on your CV, but rather what mustn't, since that seems to be the more common problem (judging by the ones I see). What I'm about to describe is based on my experience at a U.S. research lab; I imagine it generalizes quite a bit beyond that, but I can't say how far, and it is certainly country-specific.
I'll discuss five rules below, with some overlap between them. Of course these rules are not absolute (except for the last one), but you certainly shouldn't break them without thinking carefully about it and deciding there's a good reason to do so.
(1) Your CV should represent you as a professional mathematician. Anything that is not relevant to your professional life should be left out. For example, you should generally not describe non-math-related summer or part-time jobs, hobbies, side interests outside of mathematics and related fields, etc. If there's something unusually interesting or impressive (you published a novel or are a chess champion) or that displays relevant skills (you write free software in your spare time), it's OK to mention it, but just briefly and not in a prominent position.
I've seen some hair-raising violations of this rule, in which applicants devoted considerable space to things that have nothing to do with working as a mathematician. Nobody is going to reject your application just because you put something weird in your CV, but it's not good for your image as a professional.
(2) Your CV shouldn't include anything unless you think the search committee might need or want to know it. For example, contact information is valuable, as is anything that can legitimately help judge your application. However, in the U.S. you should not list your age or birthdate, your marital status, information about your children, or your religion (unless you are applying to a religious institution). I realize this is common in some countries, and of course people will be understanding about that, but it comes across strangely to give people information they don't want and shouldn't be influenced by.
(3) You should try not to seem desperate to impress, particularly with awards and distinctions. Some people provide enormous lists of very minor distinctions, sometimes with no relevance to research/teaching/service (for example, a college scholarship from a local business club). Coming across as insecure can make you seem less attractive: an ambitious department wants to hire people who are marginally too good for them, not people who are trying hard to be good enough. As a rule of thumb, when you get your Ph.D. and apply for your first job, it's OK to list any substantive distinction from grad school. You can list a few undergraduate honors, but only if they are impressive (Putnam fellow or major university-wide prize, yes; random scholarship, no). You shouldn't list high school honors at all (well, just maybe an IMO medal, but be careful not to look like you consider it your proudest achievement).
(4) Be sure not to give the impression you are trying to obfuscate anything. I don't just mean you should tell the truth, but also that you should be clear and straightforward. For example, people sometimes feel bad about not having enough items to list in their publication or talk sections, and it can be tempting to reorganize the CV to try to obscure this. For example, you could replace the "publications" section with a "research" section in which you list not just publications but also talks and poster presentations, or even current/future research topics. This is a bad idea, since it can look like you are trying to make the information less accessible, and then everything on your CV will be looked at more skeptically. Instead, you want to make it easy to understand your CV and easy to see that you aren't doing anything tricky.
(5) Don't lie. Don't say a paper will appear in a journal until it has been accepted, even if you are sure it will be. Don't say a paper is submitted until it is, even if you plan to submit it by the time the committee meets. Don't call something a preprint until it is written down and ready to distribute (you can say "in preparation" before then, but many people will ignore this since it is unverifiable). Don't say you have received a fellowship or prize if you haven't. You'd think all these things go without saying, but I've seen a couple of people get caught on one of them. You really don't want to be the person who gets asked for a copy of their preprint and can't produce one.
Best Answer
I think that the difference is between talks that you are asked to give (by whatever method) and things like "contributed paper sessions", where they solicit abstracts from people and everyone gives a talk. Most AMS meetings have the latter. Another example of a "non-invited talk" would be your PhD defense.
Certainly all the things you discuss count as invited talks for the purposes of a cv.
I personally just have a single "talks" section on my cv, but I'm a little selective about what goes in it (eg I don't include expository talks I sometimes give to grad students or talks I give in our local topology seminar).