Introverted people tend to become embarrassed either when approaching someone or when a certain topic in a conversation runs out. It is often followed by apparently "losing the ability to find a new topic", resulting in silence from both sides.
How can someone prevent this situation?
Reading this question an old adage came to mind: Always leave them wanting more.
The best way to avoid the bit when the conversation runs out of steam is to leave when the conversation is at its peak. Imagine this: you're chatting with a co-worker about widgets, you're both very interested in widgets and the conversation is easy and fun. After a few minutes of good conversation, you excuse yourself saying you have a meeting, and would love to continue chatting later. Your coworker walks away thinking it was a great conversation and can't wait to chat with you again. In this example, you're sacrificing a few more minutes of good conversation to avoid the awkward silence when you both run out of things to say.
If you're in a situation where it will be difficult to walk away - i.e. on a train or waiting in line for something. You might consider wearing headphones, sunglasses, or another accessory which will make you appear less approachable. Even if you aren't listening to anything, headphones specifically will work well for sending the message that you aren't interested in chatting.
Silence isn't always "awkward." Sometimes a conversation has reached a natural end and needs a "break." Sometimes, a person will reasonably pause to think.
However if you want to break the silence, have a "reserve" topic of conversation at hand.
How can someone prevent this situation?
Never answer statements without an "and" statement. This can be by introducing more content or asking a question.
For example:
vs:
Now, clearly in practice this is more difficult than asking about favorite colors but I find that it works well. You will find this wikipedia article about the "yes, and..." improve technique interesting. The same idea applies to conversation.
If you want to continue the conversation, try to think and set another topic (could be a similar or very different one) and let the conversation flows again.
Otherwise "I would" politely end the conversation like:
It was a very nice conversation we had. Waiting for a next time.
or some similar in these lines.
I think also: let the other person(s) take the word. It is not mandatory that you guide the conversation, instead, let others speak too. This would help when more than two persons are in conversation.
I think there's one very important thing that others have missed. In my life, I've found this is the answer 99% of the time. I'm very passionate about sharing this thought.
I have a rule: when in doubt, turn the conversation back on the other person or mirror their statements.
Them: "I can't believe FC Dynamo Kiev lost their last match"
You: "Me either, that last goal was embarrassing" then, turn it back on them "Where did you watch the game at?" or "Can you remember a worse loss?" **
Conversations aren't just statements back and forth, it's about two people interacting emotionally. What emotion are they expressing during your conversation? Are they bored, angry, upset, sleepy? Figure out that emotion, and steer your own statements in that direction.
Does the person want to keep talking? Are they saying few words? Are they facing away from you? They might not want to talk anymore, as the other answers said. They might be tired and want you to do more talking. Do they like the topic you're on? Do they show more interest when you mention a different topic? Are they focused on something? A picture, their watch, the sky? What might they be thinking about?
Keeping your mind externally focused makes conversation easier More info on that
Take the Tangents
Often during a conversation part of it will remind you of something only tangentially related. Instead of saving the slightly related thing for later (and probably forgetting about it) follow the tangent and then when that runs out you can easily return to the main topic of conversation... The more you split off the less likely you'll run out of topics to talk about, and worst case scenario you can also resort to "What were we talking about before (first tangent you can remember)."
For example, say you're talking about a TV show (say Game of Thrones) with your friend, a mention of dragons reminds you of the Anne McCaffrey books you read a few years back. Segue into talking about those (maybe whether or not they'd make a good TV show). If you can find a new tangent before your return to talking about GoT all the better. That way if the conversation lulls you can ask "What were we talking about before the Pern books?" and hopefully revive the conversation.
Think of it as companionable silence.
I have a very good, very introverted friend and when I showed this question to him, his (only half joking) comment was: "The first mistake was to start a conversation in the first place." So I would recommend trying to re-frame the problem as "I managed 10 min of conversation, yay!", rather than "Oh no, I didn't even manage to hold a conversation through lunch." People who you meet regularly will know that you are an introvert and will not expect you to be the life and soul of the party. For them these silences will not be awkward, they will be expected.
If you find yourself in a situation where you feel that the silence is becoming really awkward, try addressing it head on: "Sorry, I can't really think of anything to talk about right now." Add a socially acceptable excuse e.g. being really tired. This lets the other person know that you don't consider it their fault that the conversation has stopped. On the other hand they then might ask you why you are tired in which case you should have a good story to hand, and alas you are back to talking. Or they might tell you that they are actually enjoying a bit of peace and quiet.
For occasions where you have to spend a long time with people you don't know you all that well (travel for work or similar) and it is not in a purely social setting ("train journey" as opposed to "down the pub"), bring reading material. Actual newspapers/magazines work better than any e-stuff, as you can ask the person you are with if they wanted to borrow one of yours (and therefore generate more potential conversation topics. Maybe.)
I'm not a big fan of the Dale Carnegie method. I don't always catch people doing this in the act, but it can make conversations feel very "off" - no-one cares about minute details of my life and I know this. There is a line between "being interested" and "interrogation" and if conversation doesn't come to you naturally it's too easy to fall into the "interrogation" trap. Having a couple (as in 2 or 3) standard questions for any situations in mind is fine (at conferences I use "Which topic are you working on ?" and "When did you arrive ?" - this usually elicits a bunch of travel stories), but leave it at that.
Everyone has things they like to talk about, if you naturally take an interest in what people like then you will find the conversation swells a lot faster. A lot of people when other people talk just think about what they are going to say next, if you rather listen attentively you can much easier give meaningful responses that can better foster conversation.
I agree with what Alex Common mentioned but wanted to add practical advice. Anyway, I read How to Win Friends and Influence People not too long ago by Dale Carnegie. Essentially the crux of one of his points was that people are generally interested in the things they're interested in. Anecdotally I've found this to be pretty accurate, especially in job interviews.
If you know them, ask them about something you know they enjoy doing or talking about.
If you don't, ask them how their weekend was. If you're past Tuesday, ask them if they have any plans for the day or maybe for after they get off work. Or maybe say "hey, I like your shoes, where did you get them?" If they spend a lot of time talking about a particular subject, keep asking them about the subject or provide some sort of input about what you know about it. Just keep asking questions and throwing in personal stories when they run out of things to say. Ask about esoteric things to avoid the small talk feedback loop. People have lots of generic things to say about the weather, but not about makes of cars.
If it seems impossible, they may just not be in the mood to talk, in which case you just need to cope. If they're in that state though they probably don't feel awkward about the silence, so it's all in your head at that point. I usually abandon transcending small talk if I can't get them yammering within 4 sentences or less.
