Castells are a world-famous tradition of building human towers:
I am going to visit Barcelona this July and would like to see one of these towers. Where and when can I do so?
Bonus points for providing a tip on how to participate in the construction of a "tower", at least in a minor role.
Castells go normally attached with festivities around Catalonia. Since these occur in summer, you are definitely going to have the opportunity to see them. People doing this go throughout cities and they even have competitions. The most important groups are from cities in Tarragona, like for example Vilafranca del Penedès and Valls (full ranking from this year). If you are lucky enough, you may find them in Barcelona.
In the webpage of the Coordinadora de Colles Castelleres de Catalunya (an Association that coordinates the activities of the groups of castells in Catalonia) you can find the calendar of performances: http://www.cccc.cat/contents/july-51 Just go through it and check the city that is more convenient for you.
I would also suggest you to check the calendar of the Colla from Barcelona, since they are the most likely to be showing in that city. Their calendar is in http://www.castellersdebarcelona.cat/calendari/ and, for example, I see they have a performance by 16th July in the neighbourhood of Raval (Chinatown).
