I just walked by a Korean Church that had very Chinese-looking writing on it. How does modern written Korean relate to Chinese? Is it a simplified version that replaces brush strokes with lines? Does older Korean look more brush-like?
Korean uses a phonetic alphabet. It looks like there are more symbols than there are because the basic letters are grouped together to form syllables rather than being written in a linear fashion. The Hangul:
Traditionally, Korean was written with both the 한글 (Han-Geul) and 한문/한자 (Han-Mun/Han-Ja) writing systems. The Han-Geul is the phonetic (really phonemic) alphabet and is written with one syllable together so that it takes up the space that one typewritten English letter would. Han-Mun are Chinese characters and traditionally were used to write the base words, with the grammatical particles (marking for tense, case, etc.) written with the Han-Geul.
After the establishment of the two separate nations in Korea, the People’s Democratic Republic of Korea (North Korea) completely phased out using the Han-Mun. The Republic of Korea (South Korea) goes through phases of using the Han-Mun or not using it. Currently, South Korea is in the not using Han-Mun phase.
When I first learned Korean, back in the late 1970s, one had to learn the Han-Mun. Now, very few people are proficient with it.
Thanks all
All vs. both brings up a different linguistic issue, Dan. 
To add some anecdote to the excellent information provided by all 
of Monty and yabob, the Korean alphabet is actually quite easy to learn to read. The shapes follow a pattern and it’s easy to distinguish vowels from consonants. Having each syllable in a separate block simplifies things as well.
Younger (under 30) Koreans have very poor Chinese character reading skills these days. It’s somewhat of a social concern, because the vast majority of historical Korean writings (documents, poetry, monuments) are non-alphabetic. For comparison, imagine everything written before 1900 in the US was in Latin. You can read translations, but not the originals.
Fortunately, the Chinese language is becoming more popular in Korea, and with it basic Chinese character reading skills. So character reading is likely to come back into fashion again sometime.
Is it still the custom for Koreans to write their names using Chinese Characters? Are there any other instances where this is standard practice-- famous historical landmarks, for instance?
Wait. What’s the distinction between phonetic and phonemic?
It depends. Most of the time, Korean people will write their names in Korean and not in Chinese characters.
Phonetic means writing the sounds pronounced, phonemic means writing the sounds “heard”. A simple example is the t phoneme is pronounced differently in the words tub, but, and butter. A phonetic transcription would write the t sound differently in each word. But native English speakers will interpret them all as t.
My friends and I used to drive our home-room teacher at the Vietnamese course in the Defense Language Institute up the wall. Vietnamese doesn’t have a distinction between ‘both’ and ‘all’. Basically, they say ‘all two’ or ‘all three’ or ‘all {number}’. When we would translate orally, we’d translate “ca? hai” as “all two” and “ca? ba” as “both three.” He didn’t see the humor in it, oddly enough.
For the main consonants, they’re based on the shape the organs of speech assume to make the sounds represented. And, yes, it’s pretty cool the way the syllables are written in separate blocks. Korean dictionaries alphabetize words by syllable.
Also contributing to the resurgent interest in Han-Mun is the popularity of learning Japanese, which still uses the Kanji/Kana system of writing.