At the AP level it may be best for you to focus on some examples. The mean and the median are two of many different ways to summarize data. Presumably, the person who took the trouble to collect the data had a reason for doing so, and it is important for a statistician to understand that reason.
Salaries of employees at a company. These will often have a lot of lower numbers with a few high numbers mixed in. The median might give a fairer summary of what most workers get paid. If someone is interested in how to find the money to pay everyone, then the mean would probably be best because it is derived from the total payroll. If we are interested in the impact of raising the minimum wage, we may ignore both the mean and the median and just look at the salaries of the worst paid quarter of the workers.
Breath flow of asthma patients. Such patients may be asked to blow out as much air as possible in a short time while we measure the volume. There may be a lot of really low numbers. Maybe the median is best for focusing on those. But I would want to know what the lowest numbers really mean. Do they indicate patients whose asthma is under pretty good control but just the worst of our group, or do they indicate people who are in really serious trouble and need emergency attention? If I'm wondering whether these patients are breathing more freely than another group we saw last week, I might want to compare the means of the two groups.
Earthquake magnitudes. You can go onto a website and get the magnitudes of all earthquakes in California for the last month. There will be a lot of very low numbers corresponding to seismic events detectable only by very sensitive instruments. (Normally, values below about 1.5 on the Richter scale are ignored because they may have been mining explosions, construction noise, etc.) There may be a few quakes around 3 or 4 on the scale, that were felt only by a few people near the epicenter and that did no real damage. If there happens to be one quake with magnitude 6.5, that may the one and only number out of a few hundred that is of any importance to the general public. For most people this outlier is the only number that matters.
Ordinal scale data. Suppose you have data from a poll asking whether people favor a tax on sugary sodas to pay for after school athletics. Possible responses might be '1=Strongly oppose', '2=Oppose', '3=Neutral', '4=Favor', and '5=Strongly Favor'. Because these opinions can be sorted in order from most opposed to most in favor, we could pick out the middle opinion in the ordered list and say that is the median opinion. It is questionable whether the mean would be of any use. (The numbers are just substitute labels and can't really be added, so there is no true sum and no real mean.) Maybe the mode, the most common opinion would be of interest.
Grade point averages can be difficult to interpret for similar reasons. Grades F, F, A, A are hardly the same thing as four C's. Historically, GPAs came to be used because averages of many 'numbers' are easier to compute than medians, and computation of GPMs (Grade Point Medians) would have been impossible before modern computers. You can think about whether GPMs might be better than GPAs.
Measures of Dispersion. Sometimes measures of dispersion such as the standard deviation are important in addition to measures of centrality such as the mean and median. The mean annual temperature at some places in the Midwestern US are the same as in the San Francisco Bay Area. But the dispersion of temperatures is markedly larger in the Midwest. If you get a mean (or median) of 55 from numbers between 35 and 85 (Fahrenheit), it's a lot different from getting a mean (or median) of 55 from numbers that range from -20 to 110.
Understand first, summarize second. Nowadays statisticians are dealing with larger and larger datasets. In the future we will undoubtedly need descriptive statistics other than the ones that can be discussed in AP statistics classes.
I don't mean to ignore 'properties' of the mean and median. It is worthwhile
noting that if you have means from two groups of 20 you can find the mean of all 40, but this doesn't work for medians. But the important starting place is to understand the purpose and meaning of the data and then use descriptive statistics (and graphs) that honestly convey that meaning.
I know it is common in elementary statistics courses--especially ones taught
by people who have done little real data analysis--to refer to the mean, median, standard deviation, etc. as "functions." But that word focuses attention on the mechanics. Maybe it's more useful to think of these numbers as "descriptions."