Unfortunately, the term "random sampling" is sometimes used to mean haphazard or careless selection.  It is anything but.

Social researchers speak more commonly of probability sampling, meaning that every member of a population being sampled has a known, non-zero chance of being selected into the sample.  Usually, the probability of selection is also less than 100%, so random selection methods become involved.

In the simplest case, random selection means that each member of the population has the same chance of being selected, and the chance of being selected is not affected by who else is selected.  Therefore, if your best friend or your mother were selected into a random sample, your own chance of being selected is no better or worse because of that.

Flipping a coin or rolling a die is a good, physical illustration of random selection.  When you flip a coin, there's a 50/50 chance of getting heads.  If you flip it once and get heads, the second time you flip the coin, there's a 50/50 chance of getting heads.  If you flip the coin 100 times and get heads each time, there is still a 50/50 chance of heads on the 101st flip, though you might want to inspect the coin carefully in that case.  That's the bottom line of a random selection process: the equal probability and independence of events.

There are several types of probability sampling designs, based on this principle.  The simple random sample or Equal Probability of Selection Method (EPSEM) is the most straightforward.  If we had a list of everyone in the population we were interested in studying, they could be numbered from 1 to n (the total size of the population).  If we wanted to select a random sample of 400 people, we could select 400 random numbers (either pick them from a table of random numbers or have a computer generate the numbers for us) and the 400 population members who had those numbers would be in the sample.  It's like the lottery.

Probability theory tells us that if a sample is selected from a population in such a way that every member of the population has the same chance of selection, then the sample that is selected will be representative of the whole population.  That is, anything we learn about the sample (e.g., the percent female, the average income, etc.) will closely approximate what we would have gotten if we had been able to study everyone in the whole population.  The larger the sample selected, the more representative it will be.  The textbook discusses this in terms of sampling error.

Though the simple random sample is the elementary probability design, it is not necessarily the most common.  Often researchers select a systematic sample, wherein kth person (e.g., every 10th, every 233rd, etc.) in the list is selected for the sample.  (We say that k is the sampling interval.)  Suppose there are 100,000 people in the population, and we want a sample of 1,000.  We could accomplish this by selecting every 100th person.  Unless the list was structured in such a way that every 100th person was a special sort, this procedure produces the same result as simple random sampling.  (In one military study, the researchers selected every 10th soldier, not realizing that the list was organized by squads, and every 10 soldier was a sargeant.  Usually this is not a problem.)

Whenever a systematic sample is used, you should pick the first person at random: select a random number between 1 and the sampling interval.  If you are picking every 900th person, pick a random number from 1 to 900, say 246.  Then your sample would be people with the numbers: 246: 1146, 2046, etc.

I've said that random sampling produces samples that are representative of the populations from which they are drawn, and the larger the samples, the more representative they are.  That representativeness can be increased by stratified sampling.  This means organizing the list into homogeneous groups or strata before selecting a systematic sample.

Consider this.  If you select a simple random sample from a college student body, you will get about the right numnbers of men and women, approximately the same percentages that exist in the whole student body.  The percentages are unlikely to be perfect, however.  You are likely to have a few too many women or not quite enough.  This what's meant by sampling error, and the aim is to reduce it.  Imaagine that we rearrange the list of students so that all the men come first, then all the women.  Now we take a systematic sample through the list.  This time, we should get exactly the right numbers of men and women, or be off by one person at worst.

We could also select a stratified, random sample.  In this case we would group the student names by gender and pick the same proportion of each, using random selection within each subgroup.  If the population had half men and half women, we would pick the same number of each in our sample.

The final sampling design I'll introduce here is the cluster sample, sometimes called a multistage cluster sample.  This design grows out of the fact that the people we want to sample often cannot be found in a single list.  Imagine that you want to select a sample of college students in the United States.  There is no list of the nation's student body.  However, there is a list of colleges and each of those colleges has a list of its students.  The colleges would be the "clusters" in this case.  We'd begin by selecting a sample of colleges (random or systematic, stratified or not).  This would be the first stage of sampling.

Then, we would obtain lists of students from each of the selected colleges and draw separate samples from each of those lists.  The final result would be a sample of college students that represented all college students in the US.

The textbook discusses each of the sample designs in greater detail and depth.