Many aspects of Eukaryotic Replication are similar to Prokaryotic Replication. Therefore, make sure that you are familiar with the process in E. coli. What is decried here are just aspects that are either unique to eukaryotes or which differ from what we find in prokaryotes.
For this section you should understand the following aspects of Eukaryotic Replication:
Initiation: The main difference is that initiation occurs at multiple sites along each Eukaryotic chromosome. These chromosomes are usually much larger than prokaryotic chromosomes so time constraints alone prevent single origin. In yeast, these sites are referred to as ARS (Autonomously Replicating Sequences) and they have the consensus sequence ATTTATYTTTA - obviously A+T-rich. Replication is bidirectional from each origin.
Elongation: While E. coli has a single DNA Polymerase that replicates the chromosome, eukaryotes have multiple DNA Polymerases that are involved. DNA Pol alpha is tightly associated with a primase enzyme and is required at Initiation and for the priming of each Okazaki fragment so it is involved in lagging strand synthesis. DNA Polymerase delta and epsilon are responsible for polymerization on both the leading and lagging strands (since alpha is only involved in priming for the lagging strand) but it is not entirely clear which polymerase is responsible for which strand.
The protein PCNA acts to clamp DNA polymerases to the DNA molecule, similar to the function of the Beta clamp in E. coli.
None of these Eukaryotic DNA polymerases has exonuclease activity, which is what is required to degrade the RNA primer before it can be replaced by DNA (as in E. coli DNA PolI). Instead, ribonuclease H1 and ribonuclease FEN-1 cut out the RNA primer and it is the replaced by DNA Pol delta.
Nucleosomes: One issue that is not a factor in prokaryotes is the need to replicated DNA that is wound in nucleosomes. In Eukaryotes, nucleosomes are disassembled just ahead of the replication fork and then reassembled immediately after it passes. Of course there are twice as many nucleosome by that time so a lot of new histone proteins are required. Protein Nap-1 brings histones from the cytoplasm - cells undergo a burst of histone synthesis prior to replication - and protein CAF-1 then brings the histones to the chromosomal site where they need to be assembled. It is actually through binding to PCNA - the clamp protein - that CAF-1 is able to bring the histones to the assembly point.
Telomerase: A major process that is unique to eukaryotic replication is due to the unique structure of telomeres. Telomeres have a single stranded overhang of a short repeat (TTAGGG in vertebrates). This sequence is also found in repeats within the double stranded section at the end. Most importantly, we need to recognize the physical problem of replicating chromosome ends. Since DNA Polymerases can only synthesize 5' to 3' and need to extend a primer the lagging strand can never be fully replicated: there is no template available to synthesize a primer that could be extended to replicate the extreme 3' end (see the video below). As a result, the end of the chromosome gets a bit shorter with each replication and this adds up to a lot over the thousands of cell divisions that can occur during the development of an individual! Unless something is done to counter this the chromosome would eventually disappear.
Eukaryotic cells get around this by using an enzyme called telomerase to replicate the chromosome ends. This enzyme acts to lengthen the telomere by adding numerous copies of the repeat sequence. Telomerase consists of protein and RNA; the RNA portion acts as a template which the protein can utilize to extend the 3' end of the chromosome. The template is actually what defines the repeat sequence: in humans the RNA template is 3'CCCTAA so the enzyme synthesizes TTAGGG. If this template sequence were to change (by mutation) it would result in a change in the telomere repeat sequence.
Once telomerase has added repeats onto the 3' end this can now be used to extend the 5' end by serving as a template for basic polymerase activity, although the cell always leaves a single stranded overhang.
The following video gives a nice presentation of telomere replication (for a species with the repeat sequence TTGGGG):