Picking up a jar of Bomb Paste upgrades your Bombs into 'Sticky Bombs'.
Bomb paste appears as a white jar with a red "P" on it, and is sold in bomb shops or can be found randomly within Crates. Additionally, a guaranteed Sticky Bomb upgrade can be found by killing a Giant Spider.
Once you have obtained sticky bombs, the bomb icon on the top of the screen will change. You do not run out of paste however many bombs you use, and the upgrade is irreversible.
Sticky bombs work exactly the same as regular bombs, with the same explosive yield and the same fuse length.
The difference is that sticky bombs will stick to whatever surface they touch first, allowing you to easily destroy tiles in hard-to-reach places. Sticky bombs will also stick to Damsels, Shopkeepers, and most enemies as well, making it much easier to kill moving targets with them.
Using this feature, you can turn a Damsel into a walking explosive, allowing you to bomb areas you might otherwise have had to take damage to destroy.
It's possible to plant a sticky bomb on a Damsel by hitting the bomb hotkey while carrying her, which is easy to do accidentally. You can also drop a sticky bomb onto a stunned Shopkeeper in the brief time they're down after being jumped on.
Once you pick up a jar of bomb paste, you cannot use regular bombs again until you die.
This can be inconvenient because regular bombs have various uses that can't be pulled off well with sticky bombs, such as bouncing and rolling.
For instance, you can't drop sticky bombs down a shaft or bounce them into places not within a direct arc from your position, as they often stick to walls before they reach their intended destination. If they are mis-thrown, they can't be picked up and thrown again.
In light of the above, for a player experienced with cooking bombs and throwing trajectories, it may be wiser to avoid sticky bomb pickups so that you can continue to use the bouncing of bombs to your advantage, although it is generally a better idea to collect bomb paste when you see it.