LMFDB
The L-functions and Modular Forms Database is an extensive online database of mathematical objects arising in number theory, including number fields, elliptic curves, modular forms, L-functions and more.
SageMathCloud
Sage is an open-source computer algebra system that includes extensive support for number theory. SageMathCloud provides a notebook interface to sage that runs in any browser and is hosted on an external computing cluster; no software installation is required and there is no cost to register for an account. You can also download a copy of Sage to run on your own machine if you wish.
PARI / GP
PARI / GP is an open-source computer algebra system designed for fast computations in number theory. PARI is a fast C-library and GP is a script language interface to PARI that runs in an interactive shell (command-line only, no browser interface). Many Sage functions are implemented using PARI (you can directly call PARI and GP functions from within sage).
Magma
Magma is a closed-source computer algebra system focused on number theory, group theory, and algebraic geometry. There is a handy online calculator for quick calculations, and a command line interface that you can run from many of the math department servers (type magma at the command line).