What alternate DNS servers can I use ?

Since many connectivity issues are eventually traced to DNS problems, you can add to, or even replace your ISP's nameservers with other free public ones. Your ISP automatically assigns their default DNS servers when your computer, or router connects to the internet. The problem is, their DNS servers can sometimes be congested, and cause connectivity issues, catching problems, or delays in name resolution. There are also other more subtle concerns about tracking your DNS queries and using/selling that aggregate data, so it may be a good idea to make an educated choice of who you want to trust with your browsing habits in general. Finally, some servers filter results deemed "malicious" or "unsafe", so can decide on a DNS provider that provides some type of safety net.

Below is a list of the better free public DNS servers you can use instead of the ones assigned by your ISP. Just configure your network adapter (or your router) to use the following IPs as your DNS servers:

NTT / Verio (global backbone and web hosting provider, fast DNS resolvers in the US)
129.250.35.250
129.250.35.251

Level3 (Tier 1 carrier, may filter "bad" domains)
209.244.0.3
209.244.0.4
Alternative IPs: 4.2.2.1, 4.2.2.2, 4.2.2.3, 4.2.2.4, 4.2.2.5, 4.2.2.6

Google (may filter "bad" domains and collect aggregate data)
8.8.8.8
8.8.4.4
IPv6: 2001:4860:4860::8888 and 2001:4860:4860::8844

CloudFlare (privacy oriented, no records kept. New for 2018, may have issues with some ISPs blocking the 1.0.0.0/8 range)
1.1.1.1
1.0.0.1

OpenDNS (25 datacenters worldwide, owned by Cisco since 2015)
208.67.220.220
208.67.220.222
Alternative IPs: 208.67.220.123, 208.67.222.123, 208.67.222.220, 208.67.222.222

UltraDNS - Neustar (cloud-based, DDoS protection, may filter "bad" domains)
156.154.70.1
156.154.71.1
Alternate IPs: 156.154.70.22, 156.154.70.25, 156.154.71.22, 156.154.71.25, 198.153.192.1, 198.153.194.1

Dyn (acquired by Oracle in 2016)
216.146.35.35
216.146.36.36

UncensoredDNS (Denmark, just two servers, no data collection, dns redirects, or censoring results)
91.239.100.100
89.233.43.71
IPv6: 2001:67c:28a4:: and 2a01:3a0:53:53::

Verisign (claims to respect DNS privacy, may filter "bad" domains for security purposes)
64.6.64.6
64.6.65.6
IPv6: 2620:74:1b::1:1 and 2620:74:1c::2:2

Comodo Secure DNS (15 locations, malware domain filtering)
8.26.56.26
8.20.247.20

Norton ConnectSafe (may filter "bad" domains)
199.85.126.10
199.85.127.10

Comcast
75.75.75.75
75.75.76.76

AT&T
68.94.156.1
68.94.157.1

Sprint
144.228.254.10, 144.228.255.10, 206.228.179.10 (USA)
217.147.96.130, 217.149.32.130, 213.206.128.130 (Europe)
203.222.32.130, 203.222.36.130, 203.222.38.130 (Asia)


Some of the free DNS services above may filter DNS results to keep you "safe" from malicious sites, phishing, even scam/adult content and may use aggregate data collected from you (Google, OpenDNS, Norton ConnectSafe) while others will not alter results to keep you "safe" (UncensoredDNS), and most likely not log your queries (Comodo, Verisign).

There are other alternatives to the US government DNS root zone, which cover some new TLD listings that many nameservers don't yet include. Check here for more information: Alternative DNS root.

Notes:
It is a good idea to choose a DNS server that is geographically close to you, or ones with global presence.
You can choose to replace both, only one, or add to your ISP's DNS servers. Under Windows, the "Advanced -> DNS" tab allows to specify more than two DNS servers, and the order of DNS servers used for resolution.