"En el fondo" means "on a matter of substance" or "on substantive grounds" (as opposed to "en la forma" which is "on procedural grounds"). It goes with "recurso de casación" and can just be translated as "substantive".

In British English you bring an appeal; in American English you file it. This is what "deducir" means.

Since 1999 (I think), the "demandante" in a civil case, which used to be called the "plaintiff", has been called the "claimant" in England and Wales. It's still called the plaintiff in the US, however.

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Note added at 42 mins (2013-06-04 16:39:57 GMT)
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"Substantive
Definition - adj
1 : of or relating to a matter of substance as opposed to form or procedure <a ~ issue> <the ~ instructions to the jury> <was dismissed on procedural and ~ grounds>
compare procedural"
http://research.lawyers.com/glossary/substantive.html

"The substantive appeal concerns the validity of an order made by the Treasury in 2009, which significantly and detrimentally affected the Bank’s business."
http://www.supremecourt.gov.uk/news/bank-mellat-v-hm-treasur...

"In most cases, to bring an appeal to The Supreme Court, you must first apply to the court which handed down the judgment to ask for permission to appeal."
http://www.supremecourt.gov.uk/procedures/how-to-appeal.html

"On 26 April 1999, the civil courts of the English legal system are to undergo a huge change. [...]
For example, plaintiff goes, to be replaced by claimant.
http://www.worldwidewords.org/articles/probono.htm

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Note added at 46 mins (2013-06-04 16:43:43 GMT)
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"deducir
4. tr. Der. Dicho de las partes: Alegar, presentar sus pretensiones o defensas."
http://lema.rae.es/drae/?val=deducir

"deducir acción: to file a lawsuit
[...]
deducir una demanda: to file a complaint"
Thomas L. West III, Spanish-English Dictionary of Law and Business (American English)