Solutions shown graphically:

Plot[{Tan[Sqrt[λ]],-Sqrt[λ]},{λ,0,10Pi}]

Solve does not work here but we can use FindRoot to get numerical solutions one at a time

Solve[Tan[Sqrt[λ]]==-Sqrt[λ],λ]

First eigenvalue and eigenfunction:

FindRoot[Tan[Sqrt[λ]]==-Sqrt[λ],{λ,4}]

λ[1]=λ/.%

λ[1]

Plot[Sin[Sqrt[λ[1]]x],{x,0,1}]

Second eigenvalue and eigenfunction:

FindRoot[Tan[Sqrt[λ]]==-Sqrt[λ],{λ,24}]

λ[2]=λ/.%

Plot[Sin[Sqrt[λ[2]]x],{x,0,1}]