Two different jobs
Descriptive statistics summarizes the data you have: means, medians, spreads, shapes. Its claims stop at the dataset's edge. Inferential statistics uses a sample to claim something about a population you did not measure: estimates with margins, tests with error rates. The same arithmetic can serve both jobs: a mean of 72 describes one class's quiz, and the moment you call it evidence about "students in general", you have crossed into inference and its rules.
Ten quiz scores: 58, 72, 65, 80, 91, 67, 72, 74, 69, 72. Descriptive summary of these ten: mean 72.0, median 72, sample SD 8.87, quartiles 67.5 and 73.5. Those numbers are simply true of the list. Inferential use of the same numbers: an estimate that the whole cohort's mean sits near 72, plus a confidence interval pricing the fact that another ten students would give a different mean. Description reports; inference bets, and quotes its odds.
Why the border matters
Descriptive claims cannot be wrong about the data (only miscomputed). Inferential claims can be wrong about the world even with perfect arithmetic, because the sample might be unlucky or biased. That is why inference carries the extra apparatus: standard errors, confidence levels, significance levels, power. Skipping the apparatus while keeping the population-level claim is the standard way statistics gets misused in reports and headlines.
Sample statistics dressed as population facts: "customers prefer the new design", from twelve surveyed friends of the founder. The description of the twelve may be flawless; the inference to customers in general is doing all the work, unsupported.
How the two connect
Inference is built out of description plus a model of sampling: the sample mean is a descriptive statistic, and the standard error describes how that statistic would wobble across repeated samples. Everything in the inferential half of this site (intervals, tests, p-values) is a descriptive statistic plus an honest accounting of that wobble.
Frequently asked questions
Is a census descriptive or inferential?
Descriptive. If you measured the whole population, there is nothing to infer: the population mean is a computed fact, not an estimate. This is also why census summaries divide by N, not n - 1.
Can the same number be both?
Yes. A sample mean describes the sample and estimates the population mean. The number is identical; the claims you may attach to it differ, and the inferential claim needs a standard error beside it.
Which one do exams and reports usually want?
Reports to decision-makers usually need inference (what should we expect beyond this data?) with description as the supporting cast. Audits and record-keeping are mostly description. Knowing which job you are doing is most of the craft.