What are these strings used to describe animals?
Those describe the genotype of an animal (or plant, virus etc.).

The nomenclature can be very varied and domain-specific, but for those two examples:

$dt^{sz}$ is a Syrian golden hamster model with a spontaneous mutation (i.e. occurred during breeding, without specific human intervention) which gives predisposition to seizures.
They are described in this paper:

Spontaneous seizures: a new mutation in Syrian golden hamsters. - Yoon et al. - J Hered 1976


  SPONTANEOUS motor seizures with tonic muscle spasms and severe paralytic states have been observed in inbred line BIO® 86.93 of Syrian hamsters
  (Mesocricetus auratus auratus). These animals had been inbred for 23 generations and maintained by Bio-Research Consultants. The seizures, which last from 2 to 5 hours, can be precipitated by mild stress, and occur in
  animals between the ages of 30 days and 60 days. Breeding tests have shown that the condition is transmitted as a simple recessive; accordingly, the gene symbol
  sz is proposed for this condition.


The lowercase indicates that the mutation is recessive that is, you need to have two copies of the mutation to see its effect (its phenotype).

dt comes from dystopia, another symptom of the mutation.

As for Rgs9-Cre/+;gtROSA/+ mouse, the genotype is a bit more complex.

Whenever you see /+ it means that the animal is heterozygous, that is, it only carries one copy of that specific gene.

For instance these are transgenic animals producing the bacterial protein Cre under the Rgs9 promoter. A promoter is a sequence of DNA that controls the expression of a certain gene. In this case the DNA sequence for the promoter of the gene Rgs9 was attached to the DNA sequence for Cre. This new construct was then inserted into the genome of the mouse.
The result is a mouse in which the cells that normally transcribe the Rgs9 gene also produce Cre.

This mouse line was then crossed with a reporter mouse line called gtROSA26, which expresses a certain protein called $\beta$-galactosidase in a Cre-dependent manner. 

The gene for the galactosidase is present in all the cells, but only where Cre is present (in our case the Rgs9 positive cells) it will be transcribed.
Eventually the crossing leads to the expression of $\beta$-galactosidase in Rgs9 positive cells. 
$\beta$-galactosidase has the nice property that it can change the colour of certain chemicals to blue, which allows to easily visualize the cells that produce it. This can help you answer questions like: "where is the Rgs9 gene transcribed?".

Wikipedia has a neat page on Gene nomenclature with lots of links to nomenclature guidelines.