The 737-800 is cleared to fly, and thousands fly every day.
Only the latest design revision, the 737-8 Max or "Max 8" are grounded, due to suspicion of a new automation system. (and also the Max 7 and 9).
The 737 is the most-produced jetliner in the world, with 3 major design revisions and many updates over decades. That design is extremely well refined and proven. Because of that, the newer versions like the -800 are particularly safe.
So what's all the fuss about?
The 737 itself is a refinement of earlier jets, and the "engines under wing" layout harkens back to the original production jet: the Messerschmitt 262.
Have you ever made a transition from a rear-wheel-drive car to a front-wheel-drive car? You might have noticed when you punch the gas, the car "pulls" a little to the left. You correct that with the steering wheel, and you hardly notice because you're already making steering wheel corrections all the time. It's automatic, unconscious.
The amount of that "torque steer" varies in different conditions (slow vs fast) and different cars and different engines in the same car (4-cylinder vs V-6). But nobody has to go back to driving school for it, do they?
Airplanes have things like that. Particularly, engine-under-wing aircraft have the thrust line below the center of gravity... so punching the gas will make the airplane tilt upward a bit. Just like your car's torque steer, it varies between speeds, altitudes, airplane model etc. And just like you, the pilot corrects for it at the wheel. No big deal.
Almost all jetliners are like this.
Compared to your airplane, the 737 MAX 8 has slightly bigger engines, which makes this effect a little more. And it's such a small change to the 737, that the (very reasonable) argument is that pilots shouldn't have to go back to flight school to learn how to fly the same airplane with a slightly bigger engine. Unfortunately, somebody decided this little extra bit crossed a line.
They had something of a case, because under rare conditions (high nose-up and low speed and ...) those bigger engines could cause a balance issue, making the controls feel soft, and that might confuse pilots. Imagine if you're going around a cloverleaf in your car, climbing a hill at 20 mph and suddenly you need more steering wheel than you expect. The new system (MCAS) added a bit of nose-down "trim" to help the airplane feel normal. FAA decided this was required to certify the plane.
Mind you, this part is about the MAX, not your airplane.
But when you add a whole new system, you add a whole new way for that system to fail. We don't know the answer, and accident speculation isn't what we do here, but all the evidence so far is pointing to an unexpected problem there.
Mind you, aircraft designers expect problems and design for every failure mode they can anticipate - and they did their diligence in this system too. It was designed to fail in a way that pilots already know how to fix*. Apparently this hasn't worked, and "why not" is the question of the hour.
It's all part of some very hard questions being faced in aviation about whether automation makes things better or worse. We're facing these hard problems because aviation has solved all the easy problems. They just don't have accidents the way they used to. Southwest practically proves the rule: America's safest and busiest carrier, they have had an all-737 fleet since 1971***. They have the largest 737 fleet in the world, and each plane averages 6 takeoff/landings a day.
* The up/down trim can be worked with either handwheels or pushing a switch that lets electric motors turn the handwheels for you. It's always been possible for that switch to get stuck or shorted, running the trim to the extreme position. The motors are intentionally weak and you can grab the wheels and stop/overpower them. There is also a switch to turn them off, right on the console. And that switch hasn't moved since 1967.
*** In the 1980s they leased a few 727's. But that's it.