Genetic diversity is the total amount of variation in a population (or among populations). Population structure is how those variants are distributed. For example, if you use the simple case of two alleles in a population, they may be at 0.5 Heterozygosity (which is one measure of diversity), or 50% frequency of each allele. But one allele may be fixed in one population, the two at 50/50 frequency in another, and fixed for the alternate allele in yet another. Assuming those three populations were each the same size the overall frequency of each allele would still be 50%, but the populations themselves are structured. There's a number of different ways of measuring structure, but the most common way is using hierarchical F-statistics.
The minimum number of samples to detect structure will depend on the amount of variation in the populations you're measuring, the type of data, and the specific measure of structure you're using, but a good general rule of thumb is 15-30 samples for diploid data. Obviously larger samples are more reliable, but over 30-40 the increase in accuracy will generally start to plateau. The only way to be completely sure is via modeling or doing an extinction curve..
I will just add that the way some Fstat are defined may be confusing, with for instance Fis often called inbreeding. A population can be largely inbred and with limited level of genetic structure, and therefore low Fis.
Genetic diversity is the total amount of variation in the population. Population structure is how these variants are distributed, calculating the frequency of different alleles and the frequency of genotype and try to explain this distribution.
The number has sampled it depends on the population size and the tool with which you will be studying its genetic structure. It is of the order of 50 to 100 sample if the study will be done by marqueeur morpho metric, from 30 to 50 in the case of biochemical markers, of 25 to 30 in the case of microsatellites and we can drop to 6 if the study is done by DNA chip for example 57K
Put another way: If you randomly sample two individual from a population, the probability that they share alleles is lower when genetic diversity is high. If you randomly sample two individuals from a subset of this population (e.g. from a specific geographic area) and the probability of shared alleles is higher, then you have population structure.
In practice, this can be measured in lots of ways. It is usually done with some type of Fst-statistic. For diploid, sexual system, this is some measure of differences between observed and expected heterozygosity, under an assumption of Hardy-Weinberg equilibrium. But, other methods measure structure from haplotype data.
However you do it, you are estimating the probabilities of finding shared alleles from two randomly sampled individuals. Imagine you estimated this by repeatedly sampling two individuals and then calculating the proportion of events where they shared some allele. That is your estimate of the probability. How many times do you think you would need to do this before your estimate matched the actual probability? It is sort of like asking how many rolls of two pairs of dice would it take to convince you that both pairs of dice are fair?
Just a comment: as mentioned, one useful way genetic structure can be defined is F-statistics. Fis and Fst can be seen as analogue measures, but at different levels of hierarchy, with Fst related to genes sampled within subpopulations relative to a population and Fis related to genes sample within individual genotypes relative to a subpopulation. Distribution of genes among individual genotypes can be random (no structure, HW conditions) or non-random (e.g. due to inbreeding or selection if possible) and Fis will inform about that. Genes can also be nonrandomly distributed among subpopulations (e.g. due to genetic drift) and Fst will inform about that. Thus, genetic structure can exist, quite independently, at different levels. However, genetic diversity (genetic variation) is a pre-requisite for that. There is no genetic structure if there is genetic variation. On the other hand, genetic diversity does not imply genetic structure by default...
Genetic diversity is the total number of genetic characteristics in the genetic makeup of a species. It is distinguished from genetic variability, which describes the tendency of genetic characteristics to vary.
Genetic diversity serves as a way for populations to adapt to changing environments. With more variation, it is more likely that some individuals in a population will possess variations of alleles that are suited for the environment. Those individuals are more likely to survive to produce offspring bearing that allele. The population will continue for more generations because of the success of these individuals.