Welcome to Learning Quantum.
Learning quantum computing can be daunting. The resources below are intended to provide an initial introduction into various important topics. Of course, this list is not exhaustive, and will be updated from time to time.
I recommend that you take a quick look at each link, to see how it fits your needs. Some assume a physics background, others are more oriented towards computer science, and others are more mathcentric. For the most part, I've selected resources that emphasize computing, rather than quantum mechanics or quantum communications.
Many of these links were collected by Prof. Frank Mueller for our course on Quantum Computing at NC State in Fall 2018  CSC 591/ECE 592.
If you only have time to read one thing:
 An Introduction to Quantum Computing for NonPhysicists, Eleanor Rieffel and Wolfgang Polak (2000)
Books

Quantum Computing: A Gentle Introduction, Eleanor Rieffel and Wolfgang Polak
The MIT Press, ISBN13: 9780262526678, 2014.While not as "gentle" as you might hope, this introduction is quite thorough and approachable.

Quantum Computation and Quantum Information, Michael A. Nielsen and Isaac L. Chuang
Cambridge Univ Press, ISBN13: 9781107002173, 2011.Known as "Mike and Ike," this is considered the canonical introduction to the field. Includes physics and engineering content.

Quantum Computer Science: An Introduction, David Mermin
Cambridge Univ Press, ISBN13: 9780521876582, 2007.Some of the notation is a little dated, but this is a very clear explanation of the major quantum algorithms, and the relationship between quantum and classical computation.

Quantum Algorithms via Linear Algebra: A Primer, Richard J. Lipton and Kenneth W. Regan
The MIT Press, ISBN13: 9780262028394, 2014.If you are comfortable with linear algebra, and want a mathoriented introduction, this might be a good start. If you are affiliated with NC State, it's available as an ebook through the library.

Quantum Computing for Computer Architects (2nd ed.), Tzvetan S. Metodi, Arvin I. Faruque, and Fredric T. Chong
Morgan & Claypool, Synthesis Lectures on Computer Science, 2011.A look at systemslevel issues for building scalable quantum computers.

Adiabatic Quantum Computation and Quantum Annealing: Theory and Practice,
Catherin C. McGeoch
Morgan & Claypool, Synthesis Lectures on Quantum Computing, 2014.An introduction and explanation of quantum annealing (QA), the basis for the DWave system, and the more more general topic of adiabatic quantum computing (AQC).
Videos
Online courses and video collections that introduce quantum computing.
 EdX course from Berkeley: Quantum Mechanics and Quantum Computation, 2013
 EdX course sequence from TU Delft
 EdX course sequence from MIT
 Michael Nielsen's YouTube series: Quantum Computing for the Determined
A collection of lecture notes from various university courses. Lecture Notes
 Quantum Computation by David Mermin
 Quantum Computation by Umesh Vazirani
 Quantum Computation by John Preskill
 Theory of Quantum Information and Introduction to Quantum Computing by John Watrous
 Quantum algorithms by Andrew Childs
 Quantum Computing Since Democritus by Scott Aaronson
 Quantum Computing by David Bacon
 Quantum Computation and Information by Ryan O'Donnell, John Wright
 Quantum Computing by Peter Shor
 Qubits, Quantum Mechanics, and Computers by Birgitta Whaley, Kevin Young, Mohan Sarovar
 Topics in Quantum Computing by Don Towsley and Bo Jiang with videos
 Quantum Computing by Ronald de Wolf with fullfledged lecture notes
 Great Ideas in Theoretical Computer Science by Scott Aaronson with lectures on Quantum Computing and Quantum Algorithms
Industry Research and Tools

IBM Q
 IBM Q Experience  public machines, simulators, tools, tutorials
 Qiskit  software development
 Microsoft
 DWave
 Rigetti Computing
 IonQ
Papers / Presentations / Other Tools
General
 Design of a Superconducting Quantum Computer, John Martinis
 Software and Architectures for LargeScale Quantum Computing, Frederic Chong
 Quantum advantage with shallow circuits by Sergey Bravyi, David Gosset, Robert Koenig in arXiv:1704.00690, Apr 2017. Also in Science, Vol. 362, Issue 6412, Oct. 2018, pp. 308311, slides, video
Simulators
 Quirk: a draganddrop quantum circuit simulator
 QX simulator from QuTech (TU Delft and TNO)
 See industry links above: IBM, Rigetti, Microsoft, DWave
 List of QC simulators
Compilers, Languages

ScaffCC: compiler for the Scaffold language
 ScaffCC: Scalable Compilation and Analysis of Quantum Programs, Ali JavadiAbhari, Daniel Kudrow, Jeff Heckey, Alexey Lvov, Frederic Chong, Margaret Martonosi. arXiv:1507.01901, July 2015. Also in Parallel Computing, Vol. 45, June 2015, pp. 217.
 Quipper: functional language for quantum computing
Error Correction
 Quantum Error Correction for Beginners, Simon J. Devitt, Kai Nemoto, William J. Munro. arXiv:0905.2794, June 2013. Also in Reports on Progress in Physics, Vol. 76, No. 7, June 2013.
 Surface codes: Toward practical largescale quantum computation, Austin G. Fowler, Matteo Mariantoni, John M. Martinis, Andrew N. Cleland. arXiv:1208.0928, Oct 2012. Also in Physical Review A, Vol. 86, No. 032324, Sept 2012.
Applications
 Machine Learning: Supervised learning with quantum enhanced feature spaces, Vojtech Havlicek, Antonio D. Corcoles, Kristan Temme, Aram W. Harrow, Abhinav Kandala, Jerry M. Chow, Jay M. Gambetta, arXiv:1804.11326, June 2018.
 Financial Risk Analysis: Quantum Risk Analysis, Stefan Woerner, Daniel J. Egger, arXiv:1806.06893, June 2018. Also in npj Quantum Information, Vol. 5, Article 15, Feb 2019. https://doi.org/10.1038/s4153401901306
 Chemistry: Hardwareefficient variational quantum eigensolver for small molecules and quantum magnets, Abhinav Kandala, Antonio Mezzacapo, Kristan Temme, Maika Takita, Markus Brink, Jerry M. Chow, Jay M. Gambetta, arXiv:1704.05018, Oct 2017. Also in Nature, Vol. 549, Sept 2017, pp. 242246.