HP is the thing that is commonly referred to as Health. It is a character's life in all of the Mother Games. Whenever HP runs out, that character will die, and you must find some way to bring that character back to life. The max HP is 999, and can only be obtained through hacking. When you get hit, the HP will drop. The only way to bring it up is by using a move like Lifeup, or using an item like the Nut Bread. Any healing item will work, but it varies on how much the items will bring up. Lifeup is the best thing to use in a time like this, due to it having many different stages, and the highest one will usually restore every character's HP in that party to whatever their max HP is. The only way to get your HP higher is to gain more levels. Each time, you will earn more points in each of the few Stats so that you can be more powerful in that area. One easy way to obtain a HP boost in Mother 3 would be to go to Death Desert, then go to Wan Sum Dung after beating a Dung Beetle. He'll give you 5, 10, or even 50 EXP, so that can be a lot of help. He's only available for one chapter of the game, with only one character so that can be hard. It's very easy to obtain EXP though anywhere else in the game, but that spot is probably the easiest.
People will sometimes confuse this with PP. That is the amount of magic that can be used, not the health. It is also used in every Mother Game though, and is about the same. The main difference is this is used to tell people how much more PSI they can use.
EarthBound and Mother 3 have special engines for HP, The scrolling bar, When your hit instead of your HP immediately decreasing, It will scroll down in a slow rate (This does not happpen in Mother and scrolls Much faster in EarthBound) This can be used for your advantage, For example your hit for tremendous damage but then you attack while your HP scrolls down, You defeat it while it's still scrolling but then it freezes on whatever number it was currently in. The same works for healing. You can use a food item, or a healing PSI spell while the bar is scrolling, and it will recover the HP your HP. Another feature is slowing it down, For example you are hit with damage more than your Maximum HP, It will say you took Tremendous damage, Your HP bar will scroll much faster than normal, If you Guard with that character it will scroll down slower, Then the other characters can proceed with finishing the fight. This Engine works on PP too but it does not work, You cannot keep Casting PSI while the number is still scrolling over the PP cost but is going lower than that after it stops scrolling, If it did, This game would be too easy.