Several properties were searched after the arrests
A youth, 16, and a man, 21, have been charged with conspiracy to murder and conspiring to cause public nuisance by using poisons or explosives.
The man, from Bradford, was arrested at Manchester airport last week.
He and the 16-year-old, who comes from Dewsbury, West Yorkshire, will appear at Bow Street Magistrates' Court in central London on Thursday.
BBC News understands the charges faced by the 21-year-old relate to allegations of terrorism overseas.
The man was held under terrorism laws on 5 June and the youth on 7 June.
Several properties in West Yorkshire, including an internet cafe in Bradford, were searched after the arrests.
West Yorkshire Police, MI5 and the Metropolitan Police took part in the operation.
Police say they are aware operations "of this nature" can cause public concern and have been meeting religious and civic leaders in Dewsbury and Bradford to keep them informed of events.
A spokeswoman said West Yorkshire Police "values the continuing support of civic leaders and residents and asks that communities remain united".
This is a serious and complex investigation
West Yorkshire Police
"West Yorkshire Police would like to thank the public for their continuing co-operation and understanding during the policing operation over the past week.
"The force is aware that investigations of this nature may cause disruption, which has to be balanced against the need to act quickly and in proportion to certain information and intelligence.
"This is a serious and complex investigation.
"Inquiries are expected to continue into various aspects of the investigation for some time."