Ticket Buying Options/Tips
You can always buy your tickets at the box office of the theater for the show you want to see if you live reasonably close to the theater, or want to make a last minute decision about what to see that day. You will generally not be able to buy tickets by phone from a Broadway theater box office, but it might be possible for off Broadway or local theaters. If you do want to go to the box office, pay attention to your emails (if you signed up for the various discount email newsletters) and check online. Most discounts may be used in person at the box office, so be sure to print out any specials you have for the array of shows in which you might be interested.
"How Not to Buy Theater Tickets" is a wonderful reference article posted on theatremania.com and written by a former box office staffer.
TKTS is an off-price, in person, cash only ticket booth in NYC (two locations; one in Times Square) for day-of performance tickets only. You can check the website each day for availability, but must buy the tickets in person that day. The available shows are listed on a board near the ticket windows and will represent both plays and musicals. The shows that you should not expect to see available at TKTS on any kind of regular basis are shows that fill their houses most nights (you can find these numbers on a weekly basis at Playbill or The Broadway League ; look for shows selling over 90% and they will likely not be available for discounted prices at TKTS or elsewhere - but it never hurts to look :)!).
DIRECT TICKET WEBSITES
The show will determine what website is used for ticketing; meaning you won't be comparing prices on telecharge vs. ticketmaster. If you're looking for discounts, you need to access the discount website first and get a discount code, and depending on the nature of the discount, you will still be buying your tickets through the ticket service being used for that show. You will not usually find any discounts available directly on the ticket website itself. If you don't look for discounts first, you will pay full price.
I might also suggest that you complete a profile/log in for these sites so that when you're faced with a timed situation for completing a ticket transaction, you can simply log in and the site will autofill your information (you do not have to pre-set payment information).
www.roundabouttheater.com (for Rounadabout shows)
A ticket broker is a middle-party that has purchased the tickets at face value from the theater and is now offering to sell them to you at a (usually) extremely inflated price. A broker may also act as a reseller for ticket holders that find they can't use their tickets or have extras. You will pay over face value and sometimes not know where the seats are located until you have already purchased them. Also, the price quoted for the seats may be exclusive of fees that the broker is charging, and I've seen those fees be more than double the price of the ticket. All that said, you may be able to get those "Book of Mormon" tickets you can't live without :).
If you do a search on Google for "ticket brokers," you will find an array of choices, but buyer beware :).
With ebay, you may be buying from a broker or from an individual who can't make the performance for which they have tickets (or have extras). If you're trying for a show that's sold out, assume the price you pay will be at least double or triple the face value of the tickets. If you decide to bid in a ticket auction, I would recommend those that are for paper tickets (as opposed to email tickets, which can give rise to fraud) and give yourself plenty of time to receive them and deal with any problems that may arise. Look for "Buy it Now" options if you don't want to hassle with bidding (but you will pay more). I have used ebay twice for tickets to events and it worked relatively well.
HOTEL CONCIERGE STAFF
A hotel is sometimes able to get seats to shows for you at the last minute by working with ticket brokers. If you don't have time to get your tickets in advance and can't get to the box office when you get there, you can check with the concierge at your hotel to see if they can get seats for you. However, be aware that the seats will almost surely not be at face value and, in fact, may be hundreds of dollars over face value. This may be fine with you for the convenience, and they are often good seats (but not always; despite the high price they may be too close or over on the side, etc.). I have done this twice in my life and once it worked wonderfully: the show was completely sold out in a tiny theater and I had tried for weeks in advance to get tickets; and the other time was a disaster, with the seats being less desirable and definitely not worth the money I was going to have to pay (luckily I was able to get out of it at the last minute). I have personally overheard hotel concierge staff offering premium price seats (over $200) to a show that was not selling out and for which discounts were available. So please be cautious.