I live in Queens, NYC. I am thinking of buying a bike for casual ride probably during weekends and sometime for commuting to the city.
Can anyone recommend which and what type of bike should I buy with budget of around $500?
Hybrid bikes are good for urban riding, they will often come with fatter tires that will provide a better ride on rough pavement. These tires also don't easily slide into grooves like street car tracks, storm sewer grates, etc, which can cause falls. You'll also have the flexibility to take shortcuts through parks etc if you want to.
I would have a look through your local classifieds first. Look for a bike in your price range that is in good repair. It should be reasonably free of rust, excessive grease buildup and the wheels should spin without much (or no) 'wobble'. While not a comprehensive list, things like this tend to show whether the bike has been well cared for or not. Remember to beware, riding a bike in poor repair is torture compared to a bike that has shifting that works (unless single speed) and does not fill your ears with squeaking, grinding or rubbing sounds ;).
If you can't find a suitable used bike, I'd recommend buying a lower-end name brand bike from a dedicated bike shop. This will fit your budget, and you will often receive free maintenance on your bike for a certain period on things like shifting adjustments, wheel truing, etc. Beware of bargain basement bikes from big box stores, even the more expensive ones from these stores can begin break immediately as the components are of low quality and the bikes may not be assembled correctly.
I'm going to take a stab at answering the question, with the added spin of my comment above. What you need at this point is enough information so you can go into a shop and try a few bikes, so the shop doesn't try to sell you the bike they have, rather than the bike you need:
I bike suited for commuting bikes and a bike you'd ride on the weekends aren't necessarily going to be the same type of bike.
A bike for commuting should be reasonably fast, comfortable, and the frame should have fittings for a rack. A hybrid such as Dana suggested above would work well. These are often aluminum or steel, sometimes with a carbon front fork. (I'd stay away from a carbon fork, especially on the streets of Queens.) Aluminum frames are light, but they give a harsher ride -- you'll feel every pothole! Steel is a more forgiving frame material, it soaks up a lot of rough-road vibrations; and your hands will thank you later.
You could also ride a touring bike. Those are essentially steel road bikes, designed to carry stuff and be comfortable for longer rides. These can be a little bit of overkill, and you'd be hard-pressed to find a good touring bike in the $500 range.
A bike for casual weekend riding (what I'm thinking of as a bike-path bike, whether or not you're on a bike path) would be comfortable when ridden at a slow to moderate pace. Something with a more upright posture would be called for. A large, cushy seat, large swept-back handlebars -- essentially a **cruiser* or similar bike. (I also think of these as "beach bikes" in the privacy of my own head.)
What's called a comfort bike like the Trek Navigator would also work well for weekend riding; these are bikes similar to cruisers, without the cool factor like neato paint jobs, extremely swept back handlebars, et cetera.
Cruisers and comfort bikes are good for weekend riding, but they tend to run on the heavy side, and they're not comfortable for rides of more than a few miles. Road bikes can be expensive, but they don't necessarily serve the needs of commuters. I think the recommendation of a hybrid bike may be your best bet. If possible in your budget, see if you can get something made of steel, or with some rudimentary suspension.
Suspension can be a waste of weight and it soaks up your pedaling energy and translates it into making the bike bounce up and down. It's also hard to find suspension in lower-end bikes that isn't a waste of money. You'll want to avoid full-frame suspension, no question. You can probably find a bike with a suspension fork, that is, front suspension, but try to find one that lets you turn the suspension off for when you're riding on pavement.
Another option no-one's mentioned yet is a folding bike, useful if you need to take your bike on transit during rush hour. On the other hand, spending $500 on a folding bike... eh, save your money, unless you buy used. Folding bikes cost more for the same quality of non-folding bike, usually.
If you can, bring a knowledgeable friend to the bike shop with you, and always test-ride a bike, preferable for a few miles if possible.
You'll also want to do some reading on how to get a bike that fits you; there's plenty on the web about this, but make sure your saddle isn't too low or you'll be paying for that later -- or rather, your knees will. (The bike approach to the Queensboro bridge alone will take care of that.)
In closing, you can ignore all this advice and buy the bike you enjoy and are comfortable on. What works for me won't necessarily work for you.