Astronomers like to measure distances by the amount of time it takes light to travel such distances. For example, the nearest star to our Sun – Proxima Centauri – is about 4 light years away, so light takes about 4 years to get from it to us. In miles, a light year is about 6 trillion (10^12) miles!
The nearest black holes to us are a few thousand light years away, so about 10^15 or 10^16 miles. These black holes are similar in mass to our Sun, and are not heavy enough or close enough to have any significant effect on us.
"Many other things are likely to happen before our Solar system is sucked into the super heavy black hole at the centre of the Milky Way. We could also worry about the fact that our galaxy is on a collision course with another..."
It’s believed that every galaxy has at least one, and often more than one, enormous black hole at its centre. At the centre of our own galaxy, the Milky Way, there’s a very heavy black hole which has a mass millions of times that of our Sun. It’s around 100,000 light years away, so about 10^17 miles.
All matter which is around a black hole will eventually be pulled in. The defining feature of a black hole is that anything falling into the surface can never escape once it’s passed the surface. This surface is called an event horizon, and is a point of no return. Nothing can escape from a black hole, not even light – that’s why it looks black when observed by a telescope.
However, many other things are likely to happen before our Solar system is sucked into the super heavy black hole at the centre of the galaxy. Long before this, our Sun may die, turn into a White Dwarf and either swallow the Earth or push the Earth out of the Solar system.
We could also worry about the fact that our galaxy is on a collision course with another galaxy, Andromeda. The Andromeda and the Milky Way galaxies are predicted to collide in approximately 4 billion years.
All of the above events – the death of our Sun, the collision with the neighbouring galaxy, the black hole at the galactic centre pulling our solar system into it – would happen on timescales of billions of years. This is so far into the future that it’s really not worth worrying about! We have much larger short-term concerns, such as climate change, to worry about.
Actually, the closest object, that we're 100% sure it's a black hole, is the Sgr A* - a supermassive black hole in the center of our galaxy (about 26,000 lightyears) with a mass of about 4 mln Solar masses.
The so-called stellar mass black holes (with characteristic masses of about 10 times the Solar mass) appear in the systems of the so-called x-ray binaries. It's an ordinary star that gives away its matter to the companion through the process called accretion. The problem is, that we don't see the partner. We only see the bright radiation from the infalling matter, and can thus measure the mass of the partner. But we think, that this "invisible" partner might be a black hole, so those objects are called black hole candidates.
In particular, the closest black hole candidate in the x-ray binary system is A0620-00 (just 3000 lightyears away). The binary system consists of a K-type low mass star that accretes matter on it's partner, the presumed black hole, with the mass of 9-13 Solar masses.
Again, while these candidates are most likely to be black holes, we are not 100% sure in that.
And just a couple of remarks, regarding the previous answer.
- We definitely don't see anything "black", when we look at the stellar mass black hole. In fact, we don't see anything, but the radiation of an infalling matter, or sometimes a jet (in case of supermassive black holes). There are complicated techniques to try to reconstruct the black hole silhouette from VLBI observations (see this). But they're indirect. So there are no direct observations of a black hole. Maybe the future Event Horizon Telescope will be able to shed light on this. But not yet.
- "All matter which is around a black hole will eventually be pulled in" - that's not true. The gravitational field of a black hole far from the event horizon is not different from a gravitational field of a star with the equal mass. So you can have a planet, orbiting infinitely long around a black hole, and a Sun will probably never fall into the center of the galaxy.
There is a really nice video, where the most common misconceptions about black holes are discussed.