Scott's answer is excellent and covers the functional aspects. Let me try and translate that into a richer description of a "typical day" and important kinds of "non typical" days. The substance is basically the same, but the day-to-day implications may not be obvious if you haven't done it. I am sort of blurring the lines between project and product management because they ARE blurred (if you go one level higher to things with names like "program management", then you get a very different daily life that is more like that of an investor managing a portfolio).

Typical Day

1. Get up, drink coffee, shower, put on pants one leg at a time like everyone else
2. Check email, besides the routine types of email that everybody does, PMs often write a very specific kind of email that I call a "closure, options and actions" email. You are usually managing 2-3 complex conversations at a time, that involve questions about technology, design, what users have said, what management wants, budget issues, HR issues etc. Everybody contributes. Your job isn't to do all the thinking. But the PM does do one thing others don't: notice when a conversation is reaching diminishing returns from available information. At this point you typically step in, frame and summarize what has been said, lay out the options, and either recommend or assign actions.
3. Get another cup of coffee
4. Dive into Google Reader or Twitter. Again, mostly you consume news like other people, but there are a few things you not differently that fit well into a SWOT framework (strengths, weaknesses, opportunities, threats). You filter out the irrelevant/personal interest stuff. Anything that is actionable intelligence goes into your processing stream. If it is obvious what to do, you delegate immediately (for instance, I noticed a blog post about upcoming changes in Facebook login protocols a couple of days ago, which affect our in-development FB login feature, so I just forwarded that to our lead devs). If it is not obvious, it goes into your "thinking hopper" (bookmarks, a quick scribbled note to yourself like "think about implications of Google announcement."). This goes into your Inbox. Most days come up empty, but about once every week, a major or minor item will pop up that needs handling this way. You have no option but to go fishing regularly. I do quick 20-30 minute "market situation awareness" stints every day, and longer "catch up on my Google Reader" sessions of several hours, usually on Monday mornings.
5. Lunch
6. Okay, now for something that finally looks like "real work" to some of you. Roll up your sleeves, fire up your main product management tool. It could be a team to-do list, a formal agile management tool (we use ScrumWorks) or something like Microsoft Project or even an Excel sheet, depending on what aspect of the project you are working on. Your email and paper inbox are probably full of a bunch of pending items for which the right action is to do something in your product management tool. Here, a lot of the grunt work is simply clarifying and defining items for discussion in product management meetings. This means taking an email thread that straggles around the subject "hey, maybe we oughta consider integrating with Twitter" and defining a bunch of meaningful backlog items to be done later by appropriate people. Often you don't know enough technically to actually define it yourself, in which case you might write the best stub you can for others. Sessions wrangling budgets and resources look and feel similar.
7. Whew! An hour of working with a tool like that can be exhausting. Coffee break.
8. Now, you probably have some sort of formal messaging/influence work pending. Anything from a powerpoint presentation for a review meeting to a blog post. Create/do it.
9. Wrap. Go eat dinner, chill etc.
10. Night, just before bed: attack email inbox one more time, close off as many conversations as you can. Check any inbound analytics that might have new information, such as daily traffic reports.

Typical "Unusual" Days

You will also encounter several less-common types of days. I'd say one in every 6-10 days is one of these exception days rather than the typical day above.

1. Meeting days: dominated by something like a regular weekly or biweekly project meeting. Plan on an hour of prep for every two hours of meeting time. More if you plan to do a short presentation about something. You will probably need to shoot off a bunch of preparatory emails to make sure others are going to be ready and not wasting time. You will need to prepare any unusual supporting documents. Exceptional meeting days include high-stakes reporting meetings to senior managers (or pitch days for startups). Every productive meeting day will usually have an hour or two of follow-up work: summarizing outcomes, processing follow-up actions etc.
2. Big-picture thinking days: These are days that make it all worth it. You go to your favorite coffee shop, with a bunch of blank paper. For the last several weeks or months, the low level news and project work has been creating a higher-level situation awareness in your head that something is not quite right at the 50,000 foot level. You are going to tease it out. Typically it will take 4-8 hours. You may be thinking through something like "what should our China strategy be?" or "Do we really need to worry about mobile?" or "Is our team balanced, or do we need to rethink resources?" or "Is it time to explicitly lay down some core design principles for our product and team?" or "What's the plan for our big user-testing session?" You will be looking to generate, through a few hours of scribbling, new insights into what to do next at time-frames 2x-10x your usual one (if your normal cycle is biweekly due to your regular meeting schedule, this might be a "6 months out" thinking session). You will also be figuring out how to operationalize any insights you come up with. This is not plug-and-chug. Thinking sessions like this generally lead to unpredictable output that needs custom handling. Generally, a day like this will require follow-up in the form of a presentation or Excel model or something that you share with the appropriate group in either a routine milestone meeting (like an annual kickoff meeting), or an exceptional meeting ("Special meeting on mobile strategy.")
3. Vision Think-Through Days: These are not the same as big-picture days. These are product visioning days. In software, this is a wireframe day for sketching out a completely thought-through feature idea. Unlike big-picture thinking days, you are not analyzing an abstract problem. You are creating a very tangible (if strawman) vision for people to react to. In very small teams, this might actually be an early stage design because you are doubling as the UX designer. In larger teams, this isn't a design-intent wireframe but more a "concept exploration" design, that just fleshes out all the implications of a major piece of the project/product roadmap. In mechanical engineering, the equivalent would be a day where you get to sit down and make lots of sketches of components and assemblies, even though you know it will be a hands-on engineer who will actually figure out the details, do the math etc.
4. PM Geek Days: If you are in either product or project management, chances are you actually have some technical skills that you actually enjoy exercising. So these are another kind of enjoyable day if you get a chance to indulge. It could be something like creating a nice Excel model of market demand. Or sitting down to number crunch a pile of compiled analytics data. Or running what-if planning scenarios in some sort of specialized tool.
5. People Days: Days dominated by things like hiring and staffing decisions (either as a leader or participant), interviewing and so forth. Other people days are devoted to handling motivation issues, sensitive 1:1 conversations with unhappy reports or managers etc. These are some of the most critically important, high-leverage days. Get these days right and the rest of your year will be smooth sailing. Get these wrong and you are in for no end of pain. Some PMs love this work, others hate it. I am in the middle, and lean towards "mildly dislike." These days are emotionally draining, since thinking through the people issues carefully to deal with a single "people issues" meeting or thread of "people-related" emails is very demanding. You can usually accomplish very little else on these days.
6. Customer Days: Days spent visiting or emailing or otherwise interacting with potential or actual customers, where you are doing anything from pitching and persuading to troubleshooting and providing customer support, to surveying and questioning.