Radiation damages electronics in two major ways: cumulative dose and single-event upset. Luke and Jay have already touched on the first problem, and I'll add some more comments.
Electronic circuits lack the self-repair mechanisms of biological organisms so radiation damage tends to accumulate over years. Passive components (batteries, filters, inductors, capacitors, resistors, relays, etc) tend to be much more rad-tolerant than semiconductors, although some passive components, notably batteries and capacitors, often fail for other reasons. Discrete transistors, because of their larger sizes, tend to be more resistant than integrated circuits.
Analog electronics (amplifiers, etc) generally degrades slowly; noise levels increase, gains decrease, DC offset levels drift, etc. Conservative circuit design that tolerates these slow changes can delay but not avoid the inevitable. Digital systems, on the other hand, tend to work fine until they fail. Redundancy is of limited help since everything receives pretty much the same dose and will fail at about the same time.
While most digital designers would groan at the thought of using Apollo's relay logic today, one thing you can say for it is that it was practically immune to radiation. So were Apollo's read-only core rope and read-write core memories.
Solar arrays are semiconductors, so their degradation is a major system level design consideration. You'll usually see spacecraft solar panels rated as giving so many watts at some illumination level BOL (beginning of life) and EOL (end of life), i.e., after some specified number of years. Protective UV filters ("cover slips") are invariably provided to reduce (but not eliminate) damage from that source. They stop relatively weak charged particles, but obviously cannot be as effective as several cm of metal.
Semiconductor fabrication methods can vary greatly in radiation susceptibility, and it's not always a direct function of feature size.
I haven't looked closely at this stuff for several decades so I'm not up on the current methods and dose limits. In low earth orbit you can often get away with standard commercial parts, especially if you get clever with component placement, e.g., putting the less sensitive stuff on the outside so it can shield the stuff inside. But for many spacecraft (e.g., interplanetary missions, especially to the gas giants like Jupiter) there's little choice but to pay big money for the rad-hard space-qualified stuff because you just can't do enough with shielding and system-level redundancy.