What is the proper use of a blech/plata on Shabbat?
I'll try to stick to concise practice here.
First a warning: always remove the whole pot from the heat source (blech) before stirring, adding to, subtracting from, or otherwise disturbing its contents. Messing with a food while it's on a heat source could be cooking it.
The blech is used in one of three ways:
To leave food "on the blech" from Friday afternoon
This is known as "shehiyah"; with Talmudic stoves, you had to leave a reminder not to play with the coals. If your "blech" is just a metal sheet that sits on the stovetop, it covers the flame (or electric element), serving that role. (Some rabbis prefer that it also cover the oven knobs.) If you have an electric blech that doesn't allow any heat adjustment, this might not be an issue.
Generally the food needs to be already "hot" and "cooked." There are lots more details, but that's the basic idea.
If you want hot water, you can boil it in a kettle on Friday afternoon, then put the kettle on the blech and leave it there.
To return hot, cooked food to the blech after serving it
The best way to do this is:
To warm up cold food sometime on Shabbat
You can take cold, cooked/baked, solid food (e.g. bread, or chicken without gravy) and put it on top of a pot that's already sitting on the blech. Depending on who you ask, this pot could be:
First and foremost one must be clear that a blech does not circumvent the prohibition against cooking on Shabbos (which are elaborated upon mostly in Shulchan Aruch, Orach Chaim 318).
Strictly speaking it would be permitted to place food on the fire before Shabbos and allow it to complete cooking once Shabbos had begun (which is known as Sh'hiya, placing) but the Sages were concerned that one might become impatient and stir the coals to cook the food faster so they prohibited one from doing so.
Nevertheless the prohibition was not complete and they exempted several cases, where there isn't as much concern that one will stir the coals. First and foremost sufficiently cooked food is exempt. "Sufficiently cooked" is a matter of dispute in the Gemara (and continues to be down to the late Acharonim), being either completely cooked and getting worse as time goes by or even minimally cooked (roughly 1/2 cooked). Secondly is raw food which will not cook in time for dinner anyway (so one will let it cook over night undisturbed).
Finally for partially cooked food (in between these two other standards, or food which will cook quickly) one may leave a dish on the fire from before Shabbos to let it finish cooking on Shabbos provided that one demonstrates he isn't concerned about the food cooking fast by actively diminishing the strength of the fire by removing the coals or covering them with (even a fine layer of ) ash. These acts are counter-productive to cooking the food faster so the Sages were not concerned someone who had done so would come to stir the coals.
Using a blech is the effective equivalent of a thin layer of ash.
And while under these circumstances one may leave a dish one may not return it to the fire (chazorah) without meeting several other requirements which includes that the coals are removed or covered with ash (i.e. use a blech) even if that had not been required to leave it on the fire initially.
To summarize, a blech is required to leave a partially cooked (or easily cooked) dish of food on the stove top on friday afternoon into Shabbos, or to return (fully cooked) food back to the fire on Shabbos (contingent upon a few more qualifications).
More particulars, including how these principles apply to ovens other than contemporary kitchen ovens, see Shulchan Aruch Orach Chaim 253.