I can see brush strokes on a dining table from a previous application of polyurethane.
I want to sand these and re-apply polyurethane. I do not know whether the previous application was a water-based or oil-based product.
What should I do?
The general answer to your question is yes, you can apply one over the other - with a caveat - since oil and water don't mix you'll need a sealer coat in between the two. Shellac works well because it binds to water-based and oil-based polyurethanes while keeping the two separated. Of course, wait for each coat to dry before applying the next one.
As far as your specific situation though, it sounds like you intend to sand the runs out anyway, so it wouldn't hurt to strip/sand all the way down to the bare wood and start fresh so you don't have to worry about how each layer affects the others. If you have the patience to clean everything and start from scratch, it's always nice to know what went into your finish so you can easily maintain it in the future.
Without seeing how deep the brush marks are, it is hard to advise if a simple sanding or more drastic stripping is needed. If they are fairly superficial, I'd try sanding with 150 grit paper on a DA or vibe type sander first. If it seems like it's gonna take sanding all the way to the wood to get rid of the marks, then stripping may be easier.
As far as what type of urethane to use. I usually prefer oil based because it gives a deeper coat and usually only needs 3 or 4 coats for a really nice finish. The down side of oil based is that it takes longer to dry, can smell a bit and must be applied smoothly to avoid those brush marks. I almost always use a high density foam brush for small pieces. Apply it, smooth it quickly and don't overwork it. Always sand lightly with 220 or finer paper between coats.
Water based urethanes are easy to use, clean up after and dry fast. They are very thin (watery) and tend to be much more self leveling than oil based. The big disadvantage of water based urethane is that it takes three coats to give the same depth and look as one coat of oil. I did a bar with water based for a customer once, and it took 15 coats to give the look they wanted. So I determined that water based was fine for protection and utility use, but oil is best suited for furniture and surfaces where looks and durability count.
Just did a little research: http://woodworking.about.com/od/finishing/p/polyurethane.htm
I couldn't find out whether it needs to be the same as the previous application. IMHO, I don't think it needs to be the same, unless you're trying to match how it looks with other parts of the table.
I would decide based on:
What you will want to do is to get a polyurethane stripper with brush, a pair of rubber gloves, a scraping tool, fine grit sandpaper (200), a clean cloth, and a well ventilated room.
Start by applying the polyurethane stripper liberally and evenly over the wood using the brush. Let it stand for about 5 minutes or whatever the directions instruct you to do.
Using the scraping tool, gently scrape off the polyurethane in the direction of the wood grain, taking care to remove the scrapings as you go along using a clean cloth.
Once the scraping is completed, lightly sand the wood in the direction of the grains until the surface is smooth.
You're now ready to apply the finish of your choice, whether it be polyurethane, a stain, or paint. :)