* Full DNA sequencing. This is a multi-million dollar effort taken on by large collaborations, usually to generate a reference genome sequence for a species or particular line. Typically a variety of methods are used, from 'shotgun' style illumina, 454 longer reads, and now some newer even-longer-read methods. Long reads are important for generating accurate assemblies. Generally, some tricky portions will need to be meticulously amplified and sequenced with traditional Sanger sequencing to clean up. This gives you a complete sequence, including all inter-gene sequences, repetitive portions, etc.
* RNA sequencing. This method is the core of many molecular biology labs; you take expressed sequence (i.e. RNA that has been transcribed from DNA), reverse transcribe it to DNA, and then sequence the DNA. This gives you sequence information about the expressed part of genes (this is limited, though, because during the assembly process you'll reject many reads that don't map onto the reference sequence in standard analysis) but mostly gives you an idea of the relative expression of different genes. When people talk about gene expression going up or down in certain mutants, conditions, etc., they're usually talking about RNAseq (or qPCR if only a few loci are involved).
* 'Sampling' in this case refers to chips with various DNA fragments on it. You pass the sample DNA over it, and compatible sequences hybridize together, which is then detected via florescence. These chips are designed to carry sequence fragments that vary in the population. Together, these variants form a DNA profile. The quality of this profile depends on the design of the chip. The sequences included are generally loci that are known to affect particular diseases or to be especially variable in the population for assessing ancestry.
In cost magnitude, this goes from millions to thousands to tens.
The price of a whole genome sequence has fallen dizzyingly fast, faster than any technology of which I’m aware. We’re close to the point where it’s priced like any basic blood test. Exciting times.
When you sample a genome, you're checking for one specific string. Traditionally this is done by mixing a sample of the DNA with an enzyme that breaks apart a specific DNA sequence and then measuring whether the size of the DNA molecules have changed. Take the string that I mentioned above (FOO). Suppose that I mix it with an enzyme that cleaves along "GATTACA" such that "GATTACA" becomes "GAT| |TACA". If I mix a DNA sample containing the string FOO with the enzyme I just mentioned, then the molecule would be split into: "....GAAAAGATACCCACAGAGAT| |TACAAAAC...." If I mix a DNA sample that does _not_ contain FOO with the enzyme, no change occurs.
You know when you watch CSI shows* and the DNA evidence is presented as a bunch of light and dark bands? Those are produced by mixing the suspect's DNA sample with a standard cocktail of enzymes, treating them so that the DNA molecules are electrically charged (DNA might already have a charge, not sure), placing them at one end of a container of a standard gelatin with known pore size, and applying an electric field for a standard amount of time. The cocktail of enzymes cleaves the DNA at multiple different substrings and breaks down the DNA strands into multiple smaller substrings. Different size molecules will progress through the gelatin at different rates, so you can effectively compare two people's DNA to see whether the substrings match. You _don't_ know the person's genetic sequence, just whether or not his or her's DNA contains the same substrings as the DNA of the person that you're comparing it to.
* I'm assuming those TV shows present the DNA evidence this way. I don't watch them.
Disclaimer: I'm not a biochemist by any stretch. This is based on my memory from school, so don't take my explanation as absolute truth :-)
You have described sequencing and fingerprinting. These services use SNP sampling.
Sampling has been described several places in this thread so I won’t repeat that here;
Sequencing: https://en.wikipedia.org/wiki/Whole_genome_sequencing
Microarrays: https://en.wikipedia.org/wiki/DNA_microarray
sampling is a high resolution photo with large holes in it, a full sequencing is a full photograph with varying resolution and quality etc. There are still some areas of the genome that are inaccessible even with full sequencing. Improvements are being made with long read sequencing.