Bubble in an ancient lava flow
What you're seeing is a vug in a vesicular basalt.
A vug is simply a term for a mineral-lined cavity. In this case, the host rock is a basalt. You're seeing an ancient lava flow that trapped a bubble of gases escaping from the magma as it cooled. Over time, groundwater infiltrated the bubble. As the basalt was buried, changes in the pressure, temperature, and ion concentrations in the water caused different minerals to precipitate and even change into other minerals.
Zeolites in Northwest Iceland
Given the location (northwest iceland), the minerals are probably zeolites of various sorts. Anywhere else, you'd most likely be looking at quartz, clays, calcite, or amorphous silica filling the cavity.
Northwest Iceland is known for zeolite mineralization within vugs in vesicular basalts. There are several world-class localities for zeolite minerals in the area.
Northwest Iceland has some of the oldest basalts in Iceland (they're still very young geologically, however). They've been buried and then the material above them has been eroded away, bringing them to the surface. As we'll see shortly, that's a key piece of the picture.
How did the Zeolites form?
Initially a bubble of gases escaping from a lava was trapped as the lava cooled, forming a vesicular basalt. Next, groundwater seeped into the vesicle ("frozen" bubble). Then, the basalt flow was slowly buried by many other basalt flows. Chemical reactions occured as he temperature, pressure, and water chemistry changed due to burial.
Initially, glass and olivine in the basalt near the vesicle would have broken down into smectite clays. Olivine crystalizes at high temperatures as the magma is cooling, and therefore, it's relatively unstable at low-temperature conditions. Glasses form after the lava has erupted and is cooling too rapidly for minerals to form (a glass doesn't have a crystal structure). They're often unstable because there's no regular atomic arrangement. Given time and right conditions, volcanic glasses will turn into minerals (in this specific case, mostly clays).
As the clay-lined vug (a vesicule becomes a vug once minerals begin to fill it) was buried further, the clay minerals began to become unstable due to the changing conditions. When this happens, they transform into other minerals. This is called metamorphism. Zeolite metamorphism is one of the lowest grade (lowest temperature and pressure) types, so the temperatures and pressures involved didn't affect the minerals in the basalt, only the clay minerals in the vug.
Depending on the types of clays and the pressure, temperature and water chemistry conditions, different zeolite minerals formed from the clay minerals that became unstable. In fact, you can actually constrain how deeply buried that particular piece of rock was from the minerals within the vug.
Finally, the rock above was eroded away, and this particular section was brought back to the surface, where you found it.