First off technically Blu-Ray Discs are not DVD's (Digital Versatile Disc) yes they look the same and use a similar format, but so do CD's and you don't call those DVD's. Blu-Ray Discs can also be referred to as BRD's although that isnt as common yet.
now for your questions...
"Is the Blu-ray format compatible with the DVD players I have now?"
No, A Blu-Ray disc can not play in a standard DVD player, they Blu-Ray's use a special laser that is blue/violet to read the information, where DVD's use a red laser. The blue laser is a much smaller ray of light that can read information that is compacted much smaller on the disc that the red laser can not read.
"In the article it said it only works with HD TVs. Is that true?"
No that is not true, Blu-Ray and HD-DVD are both High Definition formats intended for use with HDTV's but they will still function on a standard definition TV.
"If so, is the Blu-Ray DVD player compatible with all of the DVDs I have now?"
YES!! That is the great thing about Blu-Ray players, they can read your old DVD's, you do not need to replace your existing collection with new Blu-Ray titles, as a matter of fact they do an even better job playing them than a regular DVD player. It is called upconverting. It can increase the picture quality of a regular DVD to near high definition quality!!
You do not need to purchase a new TV or player for that matter, regular DVD's will continue to be around for years to come. But if you want to take advantage of the better picture quality and Blu-Ray features then you will need a new player, but can continue watching your old DVD's. But since you do not have an HDTV right now I would suggest on holding off for some time until you eventually need to replace your TV before investing in a Blu-Ray player. Prices are going down on both HDTV's and Blu-Ray, eventually they will be much mor eaffordable!
hope i answered your questions and did not confuse you more