If a person contracts a virus, viral conjunctivitis for example, is it possible for the individual to become "reinfected" with the exact same strain of the virus once the person has it treated and the symptoms have gone away? I am a nursing student and am fascinated with virology, but my basic microbiology class did not go very in-depth on the subject.
The nature of infectious agents is that they transmit between organisms. This means that they have reservoirs outside a single host, and hence of course encountering the exact same strain twice is possible if this strain leaves the body and at a later stage is carried back to it. This could typically happen for example with influenza or common cold viruses, spreading from one child to other children at school and later being carried back to the original child through them.
The reason why the child will not develop another illness when being exposed to the same virus again is the adaptive immune response: After the first infection, immune cells specific to a pathogen will proliferate and differentiate both into effector cells to destroy the pathogen, and into memory cells. These can mount a much stronger immune response much faster when they come in contact with the same (or a structurally very similar) pathogen, meaning that a secondary exposure usually does not result in a proper infection and illness. Of course, this mechanism doesn't work properly for anyone with a compromised immune system.
In most cases where a viral disease develops repeatedly, this is due to evasion mechanisms. For example in some viruses, those antigens which tend to be targeted by immune systems have high mutation rates. In this case the "same" virus is actually a mutated strain which causes the same illness, but "looks" different to the immune system, and so the memory cells don't respond. Again, influenza is a great example. The reason why we get flu again winter after winter, is because this virus tends to mutate over the year with relatively low infection prevalence. Then during the winter, when the conditions are favourable for flu infections, typically only a small number of strains cause epidemics - because we were maybe immune to last year's influenza, but this year's has mutated antigens which our immune systems do not recognise yet.
As discussed in @Armatus's answer, the adaptive immune response can protect against or severely limit the progression of re-infection.
Viruses have a number of ways to escape the adaptive immune response (see Murray Medical Microbiology Ch. 48). They can interfere directly with the immune system (e.g., preventing interferon action or suppressing antigen presentation), they can hide from some parts of the immune system (e.g., by spreading directly from cell to cell, or maintaining a long intracellular latent period). Some viruses are empirically less immunogenic, resulting in waning immunity. A given viral species can have a very large amount of antigenic variability, and/or a high mutation rate for important antigenic sites. Some of these escape mechanisms help maintain an initial infection and some reduce the effectiveness of a memory response on re-infection.
Viral conjunctivitis is caused by one of several serotypes of adenovirus. Generally, infection is believed to produce protective immunity against the same strain (Ch. 52, Murray Medical Microbiology), but there are many different serotypes and, perhaps more importantly key antigens are highly variable. Adenovirus uses a few other strategies to escape the adaptive immune response. The virus can enter a latent intracellular phase (avoiding the humoral response) and block a number of important effectors of cell mediated immunity with small interfering RNAs (avoiding the cell mediated response). Despite this, infection does result in seroconversion, and empirically, protection against reinfection by the same strain (Murray Ch 52 again).
Viruses that mutate quickly (particularly notorious are respiratory viruses such as the flu and common cold) quickly produce new "strains" within the same virus family. While most people should be familiar with the terms "influenza A" and "influenza B", those are merely blanket terms for a large family of viruses with thousands of strains (by surface structure). Immunity against a given strain is no guarantee of immunity to a different strain.
In the general case, reinfection with an exact same strain previously recovered from is possible if the immune system "forgets" the pathogen (which happens with some infections, hence the need for periodic booster shots) or if the person becomes immunodeficient.