Some isotopes are radioactive because their nuclei are unstable because of number and arrangement of protons and neutrons. I dont really know much about nuclear chemistry besides this, but if you need a more in depth answer maybe ask in the physics section.
Doesn't have to be a large atomic nucleus to be radioactive! Plenty of small ones, too. Carbon-14 is a small one frequently used for drug tests and dating artifacts. Tritium (hydrogen with 2 extra neutrons) is about as small as you get in the nucleus department, and is used for boosting and configuring the explosive power of modern nuclear weapons, as well as labeling proteins and so on in biological studies. Some radioactive isotopes have very long half-lives, which make any individual atom pretty stable, though a mole of even something as stable as U-238 (half-life of 4.47 billion years!) is going to make the Geiger counter take notice. 1 mole of U-238 (238 grams, only 12.5 mL in volume, a cube about 2.3 cm on a side) has an activity of about 3 megabecquerels, or 3 million atoms disintegrating and emitting something every second. That's about 4 times the natural radioactivity of a cubic meter of soil on the earth's surface. After going through one or more steps, any radioactive isotope will eventually reach a stable configuration which is not radioactive. Lead is a common final destination for radioactive elements with higher atomic numbers.