When we say something is conserved or that there is a conservation law for a given thing, we mean that the quantity of it does not change. You neither lose nor gain any of that thing.
More specifically, conservation can come in two flavours.
Something can be globally conserved. This means that the total amount of that something in the universe does not change over time. You can measure the total amount today and every day after that, you can be sure the amount hasn't changed. It may seem silly and you may ask "Well what isn't conserved then?" but most things aren't conserved. The number of humans is not a globally conserved quantity. There's more of us now than ever before. But this is a rather weak type of conservation law. Something that is globally conserved only has to maintain the same amount across the whole universe. A global conservation does not inhibit something from being destroyed from one location and simultaneously created in another.
A stronger statement is that something is locally conserved. This means that the amount of it in any localized area is constant. The only way for the amount of that something to change in any given small volume is if some of it physically moves out of the region. You can't have any popping in and out of existence, just moving from place to place.
I should mention that a global conservation should always have a corresponding local conservation. Thanks to relativity, the only way to maintain global conservation is to maintain local conservation. However, a locally conserved value does not need to be globally conserved. It's possible to have the case where something is conserved at all small scales, but on scales large enough such that they can't be considered local, they may cease to be conserved. There are a number of ways this could happen, but the first that comes to mind is if the change in quantity of something is below the limit of the Heisenberg uncertainty on all local scales but above that limit on larger, global scales. But no matter the mechanism, the point is that global implies local, but local does not imply global. A good example is energy-momentum conservation, which is conserved locally, but not globally.
A continuity equation is how we describe movement under a local conservation law. It defines how something exists at every point as it moves along and, thus, how the local conservation law is upheld. It's called "continuity equation" because it's an equation that describes the continuous existence of the quantity as it flows from one place to another.
So to sum up, saying something is conserved means the amount in a given region does not change. Saying it is locally conserved means the amount in a small region stays constant and the only way to change it is to physically move the quantity to another place. Saying it is globally conserved means the total amount of something in the universe never changes.