Ancestral Allele explanation
SNP

Let's start with the definition that has nothing to do with the rest of the question :). A Single-Nucleotide-Polymorphism (SNP) is a kind of genetic variation that you find in population. This genetic variation is defined as a variation caused by only one single nucleotide (as its name indicates it). For example if you have in the populations the two following variants at the same locus: AGCCGT and AGCTGT, then you have a SNP at the position of the C and T. This type of variation is very common. We estimate that there are about 10 millions SNPs in the human genome. We often use these SNPs as genetic markers.

Ancestral vs derived alleles

An ancestral allele or an ancestral trait (depending on whether you look at the phenotype or a t the genotype) is the trait/allele that was carried by the common ancestor of the taxon you consider. For example: as you may know the taxon Reptilia includes lizards, snakes, turtles, birds, mammals and some other lineages. In the taxon Reptilia, the scales is the ancestral state (still exist in lizards for example) and the hairs and feathers are derived states. Usually, the ancestral state is the one that carried by the most basal lineages.

Identity by state (IBS) and Identity by descent (IBD)

If two sequences are exactly identical, then we say that they are identical by state (IBS). Such identity can occur either through convergent evolution or through common ancestry.

To understand the concept of "identical by descent" (IBD) it is important to understand the concept of coalescence. A coalescent event is, when looking backward in time, an event where two sequences were actually the same sequence. Often IBD is defined relatively to a given threshold in time. If the coalescent event occurs after this event then the two sequences are not IBD. If the two sequences coalesce before this time threshold and that the two sequences are still IBS, then the two sequences are IBD

Bridging everything together

So, if you look at sequences in a given group of lineages and see that they are all exactly the same then they are identical by state. If you know that all these sequences come from a common ancestor and even they all coalesce to a single individual in the common ancestor, then all sequences are identical by descent. But some mutations may have occurred since the last common ancestor resulting in some SNPs, then all the sequences are not IBS anymore and therefore not IBD either. You can use such sequence variance in order to create a phylogeny for example.

Hope that helps.