This is a list of suggested textbooks that a student can use to learn about a topic on their own, or to supplement the text used in a class.

This list is not meant to be comprehensive. It only contains popular topics encountered in undergraduate and the first year or two of graduate study. So, if you want book recommendations for something advanced (such as C*-algebra theory or Homological Algebra), or something specific (such as Knot theory or Coding theory), you will need to look elsewhere. I have also listed some of the best books for learning a topic, with no effort to list

Under each topic I have listed some of the books I've found most useful, as well as others that are likely to be on the recommendation lists of most mathematicians.

Books in Green = Elementary, accessible with little background.

Books in Red = Difficult to read, but considered a standard and worthwhile if you can follow it.

Books in Purple = Older and considered a classic; may be difficult to read due to antiquated notation or terminology, but also contains useful material not found in newer books.

Books labeled (DOVER) are published by Dover Publications, also known as Dover Books, which primarily publishes reissues; i.e., books no longer published by their original publishers that are often, but not always, in the public domain. Dover books are very inexpensive, often in the range of $10--$20.

Undergraduate

Contemporary Abstract Algebraby Joe GallianA First Course in Abstract Algebraby Joseph J. RotmanAbstract Algebraby I.N. Herstein

Graduate

Abstract Algebraby David S. Dummit and Richard M. FooteAlgebraby Thomas W. HungerfordTopics in Algebraby I.N. HersteinAlgebraby Serge LangAlgebraby Michael ArtinAdvanced Modern Algebraby Joseph J. RotmanBasic Algebra I, Basic Algebra II, and Basic Algebra IIIby Nathan Jacobson (DOVER)Field and Galois Theoryby Patrick Morandi

Undergraduate

Elementary Analysis: The Theory of Calculusby Kenneth RossPrinciples of Mathematical Analysisby Walter RudinUnderstanding Analysisby Stephen AbbottReal Mathematical Analysisby Charles Chapman PughMetric Spacesby E.T. CopsonReal Analysisby N.L Carothers (DOVER)

Graduate

Real and Complex Analysisby Walter RudinReal Analysis: Modern Techniques and Their Applicationsby Gerald B. FollandReal Analysisby Richard F. BassThe Elements of Integration and Lebesgue Measureby Robert G. BartleMeasure Theoryby Donald L. Cohn- (Particularly good for students interested in Probability Theory)
Measure Theoryby J.L. DoobMeasure Theoryby Paul R. HalmosHarmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integralsby Elias M. SteinPrinceton Lectures in Analysisby Elias M. Stein and Rami Shakarchi

A series of four textbooks that aims to present, in an integrated manner, the core areas of analysis.Book 1: Fourier Analysis: An Introduction

Book 2: Complex Analysis

Book 3: Real Analysis: Measure Theory, Integration, and Hilbert Spaces

Book 4: Functional Analysis: Introduction to Further Topics in Analysis

Undergraduate

Complex Variables and Applicationsby James Brown and Ruel ChurchillA First Course in Complex Analysis With Applicationsby Dennis Zill and Patrick ShanahanVisual Complex Analysisby Tristan Needham

Graduate

Real and Complex Analysisby Walter RudinFunctions of One Complex Variable I, IIby John B. ConwayFunction Theory of One Complex Variableby Robert E. Greene and Steven G. KrantzFunction Theory of Several Complex Variablesby Steven G. KrantzComplex Analysisby Lars AhlforsPrinceton Lectures in Analysisby Elias M. Stein and Rami Shakarchi

A series of four textbooks that aims to present, in an integrated manner, the core areas of analysis.Book 1: Fourier Analysis: An Introduction

Book 2: Complex Analysis

Book 3: Real Analysis: Measure Theory, Integration, and Hilbert Spaces

Book 4: Functional Analysis: Introduction to Further Topics in Analysis

A Course in Functional Analysisby John B. ConwayAnalysis Nowby Gert K. PedersenElementary Functional Analysisby Barbara MacCluerIntroduction to Topology and Modern Analysisby George F. SimmonsPrinceton Lectures in Analysisby Elias M. Stein and Rami Shakarchi

A series of four textbooks that aims to present, in an integrated manner, the core areas of analysis.Book 1: Fourier Analysis: An Introduction

Book 2: Complex Analysis

Book 3: Real Analysis: Measure Theory, Integration, and Hilbert Spaces

Book 4: Functional Analysis: Introduction to Further Topics in Analysis

Undergraduate

Elementary Differential Equations and Boundary Value Problemsby William E. Boyce and Richard C. DiPrimaDifferential Equations with Applications and Historical Notesby George F. SimmonsDifferential Equations, Dynamical Systems, and an Introduction to Chaosby Morris W. Hirsch, Stephen Smale, and Robert L. Devaney

Undergraduate

Applied Partial Differential Equations with Fourier Series and Boundary Value Problemsby Richard Haberman

Graduate

Partial Differential Equationsby Lawrence C. EvansPartial Differential Equations: An Introductionby Walter A. Strauss

Undergraduate

Linear Algebraby Stephen H. Friedberg, Arnold J. Insel, and Lawrence E. SpenceLinear Algebra Done Rightby Sheldon AxlerMatrix Analysis and Applied Linear Algebraby Carl D. Meyer

Graduate

A Second Course in Linear Algebraby Stephan Ramon Garcia and Roger A. HornMatrix Analysisby by Roger A Horn and Charles R. JohnsonTopics in Matrix Analysisby Roger A. Horn and Charles R. JohnsonAdvanced Linear Algebraby Steven Roman

Advanced Undergraduate / Beginning Graduate

Topologyby James MunkresTopologyby James DugundjiTopologyby John G. Hocking and Gail S. Young (DOVER)General Topologyby John L. Kelley (The original intended title was "What every young analyst should know".) (DOVER)

Graduate

Algebraic Topologyby Allen HatcherAlgebraic Topology: An Introductionby William S. MasseyHomology Theory: An Introduction to Algebraic Topologyby James W. VickAlgebraic Topology: A First Courseby William FultonAn Introduction to Algebraic Topologyby Joseph J. RotmanDifferential Forms in Algebraic Topologyby Raoul Bott and Loring W. Tu (Combination of Algebraic and Differential Topology)

Undergraduate

Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculusby Michael SpivakDifferential Forms: A Complement to Vector Calculusby Steven H. WeintraubDifferential Forms: Theory and Practiceby Steven H. Weintraub

Graduate

Differential Topologyby Victor Guillemin and Alan PollackTopology from the Differentiable Viewpointby John Willard MilnorAn Introduction to Differentiable Manifolds and Riemannian Geometryby William M. BoothbyIntroduction to Topological Manifoldsby John LeeFoundations of Differentiable Manifolds and Lie Groupsby Frank W. WarnerManifolds and Differential Geometryby Jeffrey M. LeeDifferential Forms in Algebraic Topologyby Raoul Bott and Loring W. Tu (Combination of Algebraic and Differential Topology)

Undergraduate

Differential Geometry of Curves and Surfacesby Manfredo P. do Carmo (DOVER)

Graduate

Riemannian Geometryby Manfredo P. do CarmoAn Introduction to Differentiable Manifolds and Riemannian Geometryby William M. BoothbyA Comprehensive Introduction to Differential Geometry, Vol. 1, 2, 3, 4, 5by Michael Spivak

Elementary Number Theory

A Friendly Introduction to Number Theoryby Joseph SilvermanAn Introduction to the Theory of Numbersby G. H. Hardy, Edward M. Wright and Andrew WilesElementary Number Theoryby Gareth A. Jones and Josephine M. JonesNot Always Buried Deep: A Second Course in Elementary Number Theoryby Paul Pollack

Algebraic Number Theory

A Conversational Introduction to Algebraic Number Theory: Arithmetic Beyond Zby Paul PollackAlgebraic Number Theoryby A. Frohlich and M.J. TaylorAlgebraic Number Fieldsby Gerald J. Janusz

Analytic Number Theory

Introduction to Analytic Number Theoryby Tom M. ApostolIntroduction to Analytic and Probabilistic Number Theoryby Gerald TenenbaumA Course in Analytic Number Theoryby Marius OverholtAnalytic Number Theory: Exploring the Anatomy of Integersby Jean-marie De Koninck and Florian LucaAnalytic Number Theory: An Introductory Courseby Paul Trevier Bateman and Harold G. Diamond

Elliptic Curves

Elliptic Curves, Modular Forms, and Their L-functions =by Alvaro Lozano-RobledoThe Arithmetic of Elliptic Curvesby Joseph H. SilvermanElliptic Curvesby J.S. Milne

Undergraduate

Elementary Algebraic Geometryby Klaus HulekAlgebraic Geometry: A Problem Solving Approachby Thomas Garrity, Richard Belshoff, Lynette Boos, Ryan Brown, and Carl LienertIdeals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebraby David A. Cox, John Little, and Donal O'Shea

Graduate

Commutative Algebra: with a View Toward Algebraic Geometryby David EisenbudAlgebraic Geometryby Robin HartshornePrinciples of Algebraic Geometryby Phillip Griffiths and Joseph HarrisRational Points on Varietiesby Bjorn Poonen

Undergraduate

A Mathematical Introduction to Logicby Herbert EndertonThe Joy of Sets: Fundamentals of Contemporary Set Theoryby Keith DevlinAxiomatic Set Theoryby Patrick Suppes (DOVER)A Book of Set Theoryby Charles C. Pinter (DOVER)Computability Theoryby Rebecca Weber

Graduate

Introduction to Mathematical Logicby Elliott MendelsonLogic for Mathematiciansby A.G. HamiltonSet Theoryby Thomas Jech (Some readers suggest the 1978 version is more suitable for beginners than the Millennium Edition.)Set Theoryby Kenneth KunenGodel's Proofby Ernest Nagel and James NewmanSet Theory and the Continuum Hypothesisby Paul J. CohenComputability: An Introduction to Recursive Function Theoryby Nigel CutlandComputability: A Mathematical Sketchbookby Douglas S. Bridges

Categories for the Working Mathematicianby Saunders Mac LaneCategory Theory in Contextby Emily Riehl (DOVER)Basic Category Theoryby Tom Leinster