If you're traveling on a budget, you'll be pleased to know there's plenty of free things to do in Barcelona. Some museums have free entry every day of the week, while others are free on specific days (usually Wednesdays and Sundays). These are the best free things to do in Barcelona.
1. Gaudi's La Sagrada Familia is the most famous sight in Barcelona. Designed by Antoni Gaudi, the Roman Catholic basilica is unfinished - despite the architect being commissioned for the job in 1883! Gaudi died in 1926 and work has continued on the building to this day. Officially, the magnificent building will be finished in 2026, one hundred years after Gaudi died.
Barcelona's most famous landmark may charge admission, but viewing it from the outside is free. Personally, I don't think going inside is worth it - the museum isn't especially interesting (unless you have a big interest in architecture) and the climb to the top gives you a less-than-spectacular view of the city.
Of course, paying the entrance fee does at least help guarantee that they finish the building (over 120 years and counting, so far), but if you're on a very tight budget, you can still appreciate 90% of the building from across the road.
2. Ramblas, Boqueria market and Plaça Reial Barcelona's most famous thoroughfare is a tourist attraction in itself. The street performers are there all day, every day and for a few cents will do their little party trick for you (if you're really, really tight, you can wait until someone else pays the 10c needed to get them to do something).
Also on Las Ramblas is the Boqueria market (though the fruit and vegetables come at a price, wandering around the stalls costs nothing) and the Plaça Reial can be found just off the main street (complete with some Gaudi designed lampposts). Las Ramblas is Barcelona's most famous street. It separates the 'barrios' of La Ribera and the Gothic Quarter. You will find the famous Las Ramblas Street Performers all along the thoroughfare.
The Picasso museum is the best showcase of works by the Spanish cubist artist. It is only free on the first Sunday of the month. Be warned, the line to get in is gigantic - get there early.
The museum is also free to under-16s and to study groups (only on Wednesday afternoons). Inquire at the museum for more information. On certain days there are free guided tours of the collection, but you have to pay the normal entrance fee to see it.
Barcelona's most famous landmark may charge admission, but viewing it from the outside is free. Personally, I don't think going inside is worth it - the museum isn't especially interesting (unless you have a big interest in architecture) and the climb to the top gives you a less-than-spectacular view of the city.
Of course, paying the entrance fee does at least help guarantee that they finish the building (over 120 years and counting, so far), but if you're on a very tight budget, you can still appreciate 90% of the building from across the road.
2. Ramblas, Boqueria market and Plaça Reial Barcelona's most famous thoroughfare is a tourist attraction in itself. The street performers are there all day, every day and for a few cents will do their little party trick for you (if you're really, really tight, you can wait until someone else pays the 10c needed to get them to do something). 
3. Parc Guell is another of Gaudi's works and, like the Sagrada Familia, was never completed. It was intended as a residential area for Barcelona's rich and famous, but it didn't receive the local support required. What exists today is a fabulous park with gingerbread houses, fountains and interesting ceramic statues that will fascinate young and old alike. The small museum charges to get in, but everything else is free.
4. Picasso Museum
The Picasso museum is the best showcase of works by the Spanish cubist artist. It is only free on the first Sunday of the month. Be warned, the line to get in is gigantic - get there early. 
The museum is also free to under-16s and to study groups (only on Wednesday afternoons). Inquire at the museum for more information. On certain days there are free guided tours of the collection, but you have to pay the normal entrance fee to see it.
5. Montjuïc & the Museu Nacional D'Art de Catalunya
One of two moutains in Barcelona (the other being Tibidabo), Montjuic has a wealth of sights for those who don't mind a bit of a climb. Take a walk with a nice view of the sea, wander round the old watchtower and marvel at the Mayor's Belvedere, a collage of broken bottles and pottery by Carles Buïgas. The Museu Nacional D'Art de Catalunya is free to under-15s, over-65s and to everyone on the first Sunday of the month.
6. Hospital de Sant Pau
Designed by Catalan architect Lluís Domènech i Montaner, a contemporary of Antoní Gaudí, this fully functioning hospital is an architectural delight and is within walking distance of La Sagrada Familia, which means you don't need to even pay for any more transport. Though you might have to side step the elderly and infirm to get your photos, the vast majority of the hospital is open for you to wander around. If I ever get sick in Barcelona, I'm coming here!
8. Barceloneta Beach
It isn't the most attractive beach in the world (artificially made, rumor has it that it is half sand and half concrete) but everyone loves to spend half a day catching some rays, right? There is also a nudist beach here (a bit of a walk from the main touristy area).
Entrance to the Barcelona Cathedral is free, so you can marvel at the Romanesque cathedral from both outside and within.
Modern art museum that offers innovative exhibitions, often incorporating modern technology. Started in 1980 to provide a showcase for art considered too experimental for normal galleries.
9. Barcelona Cathedral
Entrance to the Barcelona Cathedral is free, so you can marvel at the Romanesque cathedral from both outside and within. 
10. Metronom
Modern art museum that offers innovative exhibitions, often incorporating modern technology. Started in 1980 to provide a showcase for art considered too experimental for normal galleries.
