Dark Souls itself gives very little explanation as to what some of the game mechanics are, so here's a friendly wiki page to help you get started!
Bonfires are spread throughout the world, and act similarly to the archstones from Demon's Souls. Resting at a bonfire sets your last respawn point; if you die or use a Homeward Bone, you will reappear there. At a bonfire you can do a multitude of things including reversing your hollowing, leveling up, attuning spells, and kindling the bonfire to increase the number of Estus Flasks you can carry. With the proper tools, you can also repair weapons and armor and reinforce (but not ascend) weapons. Be warned: resting at a bonfire restores all normal enemies.
Gone are the grasses of Demon's Souls—instead, you have so many Estus Flasks you can carry, whose potency and number can be affected by Fire Keeper's Souls and by how kindled the last bonfire you rested at was, respectively. More information can be found on the main Estus Flasks page.
Kindling a bonfire is done to increase the number of Estus Flasks restored when resting at a bonfire. In order to kindle a bonfire, you must be human (not hollow). Each kindling comes at the cost of one humanity, and increases the number of Estus Flasks by five.
Hollow and Human
Like Demon's Souls, Dark Souls has two states, hollowed and human. While hollowed players cannot kindle at bonfire and cannot summon other players, the state provides no impact on combat or enemy strength. You can reverse hollowing at a bonfire at the cost of one Humanity. If a person dies while human, they will revert back to their hollow form, and if they die in any form, they will lose all loose Humanity they possessed.
Souls are the both currency and experience points of the game. They can be used to buy items and reinforce and ascend weapons, and at bonfires they can be used to level up, and are gained from killing enemies and from consumable souls that you pick up. If you die, you will lose all souls in your soul pool. The only way to regain these souls is to return to the point of your death. Your bloodstain is placed where you last died, and is touched like picking up an item. If you die before reaching your bloodstain, all souls at that bloodstain are lost forever.
While online, you are able to write messages for other players to read, warning them of enemies, traps, or items ahead. You should take caution when following messages, however—some messages may have been placed to deliberately mislead players. Unlike Demon's Souls, in order to read, write, or rate messages, it's necessary to use the Orange Guidance Soapstone which can be bought from the Undead Merchant in the Undead Burg.
While human, you can summon other players to help you out as you traverse the world. Look for white soul signs on the ground and touch them in order to summon the player. While hollow, you can leave summon signs with the White Sign Soapstone which can only be found by speaking repeatedly to Solaire of Astora, who is found in the Undead Burg opposite of the Hellkite Dragon on the bridge. If a summoned player helps defeat a boss they are returned to their world with one extra humanity point. If you die while summoned, you are returned to your last bonfire rested at. If the host dies, you are returned to where you were last standing before you were summoned. While summoned, you cannot use Estus Flasks to recover HP—instead, the host must use his own flasks, which heal the entire party. You are only able to be summoned by players around your level, and as of now there are multiple unconnected servers for online, meaning that one person's summoning sign will not be seen for all available players.
Covenants are joined as a result of a player taking an oath with an NPC. You are rewarded for working with a Covenant's rules, and punished for breaking the rules. Each covenant has different levels of membership, and each covenant advances along these levels in different ways, often rewarding players for reaching new levels. If you attack a member of your covenant, you will be considered to have sinned and your name will be written in the Book of the Guilty. A sin can only be absolved by paying an exorbitant fee to Oswald of Carim, who can be found in the first bell tower after defeating the bell-gargoyles.
Dark Souls has no item burden requirement, meaning you can keep anything and everything you pick up. However, doing this will clog up your inventory and make it virtually unnavigable. You can drop your equipment, but there is always the chance that it will turn into a vagrant, an NPC which invades others' worlds in a similar manner to the black phantoms from Demon's Souls. The longer equipment stays on the ground, the more powerful the vagrant becomes, and the better loot it drops when slain.
Dark Souls does not use the black phantom method for PvP combat from Demon's Souls. Instead, if a player uses an Eye of Death, they can increase the difficulty of three other players' world by sending NPCs after them. The afflicted player can only stop the NPCs by tracking down where the Gravelord used the Eye of Death, and invading the Gravelord's world through the sign left. More information can be found on the Gravelord Servant page.