Good starting places:
Bill_Thurston_-_A_discussion_on_geometrization_-_May_7_2001
a YouTube video
The_Mystery_of_3-Manifolds_-_William_Thurston_-_Clay_Math_Institute
a YouTube video
Jessica_Purcell_-_Geometric_Techniques_in_Knot_Theory_-_IAS
a YouTube video
Craig_Hodgson_-_Hyperbolic_structures_from_ideal_triangulations
a YouTube video
Craig_Hodgson_-_Hyperbolic_structures_from_ideal_triangulations-1
a YouTube video
Ian_Agol_-_Thurstons_Vision_Fulfilled?
a YouTube video
Nathan_Dunfield_2014_Practical_computation_with_hyperbolic_3-manifolds.-.Thurston_Conference
a Cornell University video
Jeff_Weeks_2014_The_Shape_of_Space.-.Thurston_Conference
a Cornell University video
Dave_Gabai_2012_Volumes_of_Hyperbolic_3-Manifolds.-.Cornell_Topology_Festival_2012
a Cornell University video
Marc_Lackenby_-_Cusp_volumes_of_hyperbolic_alternating_knots
a YouTube video
Selling_Real_Estate_in_Hyperbolic_Space_-_Mel_Slugbate_(Colin_Adams)_[1996]
a YouTube video
Rob_Kirby_-_History_of_Low_Dimensional_Topology
a YouTube video
Geometry_Center_-_Not_Knot
a YouTube video
Dave Futer - Change in length under Dehn filling
a YouTube video

More Bill Thurston:
Bill_Thurston_-_Landau_Lectures_Part1_1995
a YouTube video
Bill_Thurston_-_Landau_Lectures_Part2_1995
a YouTube video
Bill_Thurston_-_Landau_Lectures_Part3_1995
a YouTube video
Bill_Thurston_-_Knots_to_Narnia
a YouTube video
Bill_Thurston_-_Knots_to_Narnia_in_class_part1
a YouTube video
Bill_Thurston_-_Knots_to_Narnia_in_class_part2
a YouTube video

And a whole bunch more!
Cameron Gordon - Exceptional Dehn Filling
a YouTube video
The_geometry_of_3-manifolds_-_Curt_McMullen
a YouTube video
The_evolution_of_geometric_structures_on_3-manifolds_-_Curt_McMullen_-_Clay_Math_Institute
a YouTube video
Jeff Weeks: "Shape of Space" - Aalto University MathArt Colloquium
a YouTube video
Steve_Kerckhoff_-_Collapsing_hyperbolic_structures_from_rigidity_to_flexibility_and_back
a YouTube video
Jesse_Johnson_-_From_Hyperbolic_Geometry_to_Data_Clustering_-_2015
a YouTube video
Ian_Agol_-_The_Virtual_Fibering_Conjecture_-_ICTP
a YouTube video
Volumes_of_hyperbolic_3-manifolds_-_Dave_Gabai_MSRI
an MSRI video
Rank_of_the_Fundamental_Group_and_Heegaard_Genus_of_Thick_Non-Haken_Hyperbolic_3-manifolds_-_Juan_Souto_MSRI
an MSRI video
Hyperbolic_Geometry_and_Heegaard_Splittings_-_Hossein_Namazi_MSRI
an MSRI video
Hyperbolic_Volume_and_the_Jones_Polynomial_A_Conjecture_-_Dylan Thurston_MSRI
an MSRI video
Alexander_Mednykh_-_Hyperbolic_knots_links_and_polyhedra
a YouTube video
Dave Futer - The geometry of tunnel systems
a YouTube video
Eriko_Hironaka_-_Directed_train_tracks_for_fibered_hyperbolic_manifolds
a YouTube video
The_many_aspects_of_hyperbolic_volume_-_Ruth Kellerhals_-_2017_ICTP_Seminar
a YouTube video
Visualizing_Hyperbolic_Geometry_-_Evelyn_Lamb_MAA
a YouTube video
Length_spectra_of_hyperbolic_knots_-_David_Futer_IAS
a YouTube video
Evelyn Lamb What_we_knew_about_hyperbolic_geometry_before_we_knew_about_hyperbolic_geometry_2015_MSRI
an MSRI video

An MSRI workshop:
Introductory_topics_in_Kleinian_groups_and_hyperbolic_3-manifolds_8-20-2007_Minsky
an MSRI video
Introductory_topics_in_Kleinian_groups_and_hyperbolic_3-manifolds_8-20-2007_Bromberg
an MSRI video
Introductory_topics_in_Kleinian_groups_and_hyperbolic_3-manifolds_8-20-2007_Brock
an MSRI video
Introductory_topics_in_Kleinian_groups_and_hyperbolic_3-manifolds_8-21-2007_Brock
an MSRI video
Introductory_topics_in_Kleinian_groups_and_hyperbolic_3-manifolds_8-21-2007_Bromberg
an MSRI video
Introductory_topics_in_Kleinian_groups_and_hyperbolic_3-manifolds_8-21-2007_Minsky
an MSRI video
Introductory_topics_in_Kleinian_groups_and_hyperbolic_3-manifolds_8-22-2007_Minsky
an MSRI video
Introductory_topics_in_Kleinian_groups_and_hyperbolic_3-manifolds_8-22-2007_Bromberg
an MSRI video
Introductory_topics_in_Kleinian_groups_and_hyperbolic_3-manifolds_8-22-2007_Brock
an MSRI video

More multipart talks:
Jessica Purcell: Structure of hyperbolic manifolds - Lecture 1
a YouTube video
Jessica Purcell: Structure of hyperbolic manifolds - Lecture 2
a YouTube video
Jessica Purcell: Structure of hyperbolic manifolds - Lecture 3
a YouTube video
Genevieve_Walsh_-_Introduction_to_hyperbolic_orbifolds_and_knot_commensurability_part1
a YouTube video
Genevieve_Walsh_-_Introduction_to_hyperbolic_orbifolds_and_knot_commensurability_part2
a YouTube video
Genevieve_Walsh_-_Introduction_to_hyperbolic_orbifolds_and_knot_commensurability_part3
a YouTube video
Cameron Gordon Manifolds and Dehn surgeries vol1
a YouTube video
Cameron Gordon Manifolds and Dehn surgeries vol2
a YouTube video
Cameron Gordon Manifolds and Dehn surgeries vol3 mute
a YouTube video
Cameron Gordon Manifolds and Dehn surgeries vol4
a YouTube video
Cameron Gordon Manifolds and Dehn surgeries vol5
a YouTube video

And then of course:
Alex_Zupan_-_Generalized_square_knots_and_the_4-dimensional_Poincare_Conjecture
a YouTube video
Susan Hermiller_-_Algorithms for groups of homeomorphisms
a YouTube video