One of the best healer in the game because of how focused her kit is. Barbara can freely be obtained by all players who reach Adventure Rank 18. (with free Constellation to anyone who reached AR20 before the game gets updated to version 1.1) She is a Hydro elemental character that primarily focuses on heals, which is scales with her max HP.
She heals the party when attacking enemies while her Elemental Skill is active. Or you can switch into another character while her Skill is active, for a larger single target heal.
Weapons and Artifacts
Thrilling Tales of Dragon Slayers is definitely the best weapon on Barbara when playing solo. You will mostly be switching out immediately after activating Barbara's Skill, giving the single-target heal to your DPS character. This switch will also trigger the weapon's ATK% boost, helping you stack damage. Additionally, this weapon is only a 3-star so getting Refine 5 would be fairly easy.
For multiplayer, where you won't be switching out Barbara, Prototype Malice is the best weapon. The HP% boost on the weapon helps the scaling on Barbara's skills. The passive on Prototype Malice will help Barbara activate Her Burst more often, while also providing extra party-wide heal upon Burst activation.
Maiden's Beloved set is the best set you can get to maximize healer Barbara. Both set bonuses are specifically tailored to work with healers, which is Barbara's best role. Get an HP% Cup and Hourglass, Crown can be Healing Bonus% (prefered) or Base HP%. Regarding substats, flat HP increase might be better than HP% but having both is best. Aside from that get Energy Recharge and Elem Mastery if you want.
For a budget option before building Maiden's Beloved, Exile set is easy to get from Elites and Bosses. The 2-set Energy Recharge helps Barbara get to her Burst up faster, and the 4-set Bonus will overall give more benefits when using Barbara's Burst.
Although it's not recommended to farm for it, you can replace or mix Exile with Scholar set if you have it.
Using Barbara and Chongyun's Skills together can keep enemies perpetually Frozen while Chongyun Shatters them for extra damage. In this build, you won't need to worry about Barbara's Elem Mastery because Chongyun would be doing all the proc-ing.
Barbara's constant applying of Hydro when close to the enemy ensures melee Anemo characters have an element to Swirl. This can overall increase the DPS of characters such as Anemo Traveler.
Diluc can get close to the enemy and apply Hydro from Barbara's Skill then use his Skill/Burst to trigger Vaporize. After using Diluc's Burst his Normal attacks become Pyro, so even his Normal Attacks will be augmented by Vaporize. Same goes for Pyro characters with short Skill cooldowns such as Bennett.
If your Barbara is C2+, her Skill gives a transferable 15% Hydro DMG Bonus. This buff can be given to a Hydro DPS/Burst unit such as Childe or Mona.
Speculation Xiao's Burst converts all his melee attacks into Anemo. Barbara's Skill constantly applying Hydro gives each of Xiao's hits a Hydro Swirl for extra damage. Additionally, Xiao's Burst depletes his HP over time which can be mitigated by the contant heal of Barbara's Skill.