Sn2

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I don't understand the first one, but I can explain the second one. In Sn2 reactions, a nucleophile (electron donor) attacks an electrophile (electron deficient) hence the reaction is called nucleophilic substitution because the electrophile is the substrate. In vinyl halides, the substrate is an alkene, the carbons of which are electron dense. Nucleophiles want nothing to do with alkenes because they're looking for a substrate to donate electrons to. Alkenes usually undergo electrophilic addition.
 
Aryl halides are unreactive because you're dealing with an aromatic compound. Aromatics don't do anything nucleophilic AFAIK... they react only with electrophiles.

"bicyclo compound with a halogen at the bridgehead carbon"

For example [2.2.2]bicyclohexane, where the halogen is in the middle? I'm pretty sure the halogen would be very hindered, and thus unapproachable by the nucleophile... not sure though - EDIT: Durr... it's tertiary - tertiary halides have a very low Sn2 reactivity.
 
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In case of Aryl Halides and Biocyclo compounds the ring is too bulky and it prevents the Nucleophile to effective approach and attach the carbon bearing the leaving group; thus, preventing SN2 from happening. In case of vinyl halides, the carbon bearing the halogen is sp2 hybridized; whereas, in case of SN2 reactions the carbon bearing the halogen must be sp3 hybridized. Thus, SN2 doesn't occur in case of vinyl halides.
 
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